Displaying 1-10 of 80 results found.
0, 0, 0, 0, 0, 0, 1, 3, 11, 34, 99, 279, 773, 2103, 5661, 15160, 40373, 107355, 285059, 757273, 2013177, 5361100, 14303274, 38250297, 102538714, 275597098, 742674804, 2006661720, 5436008057, 14763754746, 40196603110, 109703958381, 300091975184, 822705857129
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n<=1, n,
(add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
end:
a:= n-> b(n) -(add(b(k) *b(n-k), k=0..n)-`if`(irem(n, 2)=0,
b(n/2), 0))/2 -ceil(2^(n-4) + 2^(iquo(n-2, 2)-1)):
MATHEMATICA
b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n-1}])/(n-1)]; a[n_] := b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[ Mod[n, 2] == 0, b[n/2], 0])/2 - Ceiling[2^(n-4) + 2^(Quotient[n-2, 2] - 1)]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
1, 2, 5, 10, 22, 44, 92, 184, 376, 752, 1520, 3040, 6112, 12224, 24512, 49024, 98176, 196352, 392960, 785920, 1572352, 3144704, 6290432, 12580864, 25163776, 50327552, 100659200, 201318400, 402644992, 805289984, 1610596352, 3221192704, 6442418176, 12884836352
FORMULA
a(n) = 2^n - A005418(n). Sum of (n-1)-th row terms of triangle A136482. a(n) = 3*2^(n-2)-2^(n/2-1) for n even.
a(n) = 3*2^(n-2)-2^((n-3)/2) for n odd.
(End)
EXAMPLE
a(5) = 22 = 2^5 - A005418(5) = 32 - 10. a(5) = 22 = sum of row 5 terms of triangle A136482 = (1 + 6 + 8 + 6 + 1).
MATHEMATICA
CoefficientList[Series[(1 - x^2)/(1 - 2 x - 2 x^2 + 4 x^3), {x, 0, 40}], x] (* Michael De Vlieger, Sep 23 2016 *)
PROG
(PARI) Vec(x*(1-x)*(1+x)/((1-2*x)*(1-2*x^2)) + O(x^40)) \\ Colin Barker, Sep 23 2016 (Magma) [2^n - (2^(n - 2) + 2^(Floor(n/2) - 1)): n in [1..40]]; // G. C. Greubel, Nov 02 2018
Sequence A005418 written in base 2.
+20
1
1, 10, 11, 110, 1010, 10100, 100100, 1001000, 10001000, 100010000, 1000010000, 10000100000, 100000100000, 1000001000000, 10000001000000, 100000010000000, 1000000010000000, 10000000100000000, 100000000100000000, 1000000001000000000, 10000000001000000000, 100000000010000000000, 1000000000010000000000
FORMULA
G.f.: x*(800*x^4-99*x^2+1) / ((10*x-1)*(10*x^2-1)). - Colin Barker, Sep 23 2013
MATHEMATICA
Rest[CoefficientList[Series[x*(800*x^4 - 99*x^2 + 1)/((10*x - 1)*(10*x^2 - 1)), {x, 0, 50}], x]] (* G. C. Greubel, Sep 17 2017 *)
PROG
(PARI)
b2t(v)=sum(k=1, #v, v[#v+1-k]*10^(k-1));
(PARI) x='x+O('x^50); Vec(x*(800*x^4-99*x^2+1)/((10*x-1)*(10*x^2-1))) \\ G. C. Greubel, Sep 17 2017
Coefficients of expansion of (1-x)/(1-2*x) in powers of x.
+10
886
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592
COMMENTS
Apart from initial term, same as A000079 (powers of 2). Also the number of compositions (ordered partitions) of n. - Toby Bartels, Aug 27 2003
Number of ways of putting n unlabeled items into (any number of) labeled boxes where every box contains at least one item. Also "unimodal permutations of n items", i.e., those which rise then fall. (E.g., for three items: ABC, ACB, BCA and CBA are unimodal.) - Henry Bottomley, Jan 17 2001 Number of permutations in S_n avoiding the patterns 213 and 312. - Tuwani Albert Tshifhumulo, Apr 20 2001. More generally (see Simion and Schmidt), the number of permutations in S_n avoiding (i) the 123 and 132 patterns; (ii) the 123 and 213 patterns; (iii) the 132 and 213 patterns; (iv) the 132 and 231 patterns; (v) the 132 and 312 patterns; (vi) the 213 and 231 patterns; (vii) the 213 and 312 patterns; (viii) the 231 and 312 patterns; (ix) the 231 and 321 patterns; (x) the 312 and 321 patterns.
a(n+2) is the number of distinct Boolean functions of n variables under action of symmetric group.
Also the number of unlabeled (1+2)-free posets. - Detlef Pauly, May 25 2003
Image of the central binomial coefficients A000984 under the Riordan array ((1-x), x*(1-x)). - Paul Barry, Mar 18 2005 Binomial transform of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...); inverse binomial transform of A007051. - Philippe Deléham, Jul 04 2005 Also, number of rationals in [0, 1) whose binary expansions terminate after n bits. - Brad Chalfan, May 29 2006
Prepend A089067 with a 1, getting (1, 1, 3, 5, 13, 23, 51, ...) as polcoeff A(x); then (1, 1, 2, 4, 8, 16, ...) = A(x)/A(x^2). - Gary W. Adamson, Feb 18 2010 An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the corner squares these vectors lead to the companion sequence A094373. - Johannes W. Meijer, Aug 15 2010 Array T(m,n) = 2*T(m,n-1) + T(m-1,n):
1, 1, 2, 4, 8, 16, ... = a(n)
1, 3, 8, 20, 48, 112, ... = A001792, 1, 5, 18, 56, 160, 432, ... = A001793, 1, 7, 32, 120, 400, 1232, ... = A001794, 1, 9, 50, 220, 840, 2912, ... = A006974, followed with A006975, A006976, gives nonzero coefficients of Chebyshev polynomials of first kind A039991 = 1,
1, 0,
2, 0, -1,
4, 0, -3, 0,
8, 0, -8, 0, 1.
T(m,n) third vertical: 2*n^2, n positive ( A001105). Fourth vertical appears in Janet table even rows, last vertical ( A168342 array, A138509, rank 3, 13, = A166911)). (End) 0, 1, 2, 4, 8, 16,
1, 1, 2, 4, 8, 16, = a(n),
0, 1, 2, 4, 8, 16,
1, 1, 2, 4, 8, 16.
Number of 2-color necklaces of length 2n equal to their complemented reversal. For length 2n+1, the number is 0. - David W. Wilson, Jan 01 2012 These could be called the composition numbers (see the second comment) since the equivalent sequence for partitions is A000041, the partition numbers. - Omar E. Pol, Aug 28 2013 Number of self conjugate integer partitions with exactly n parts for n>=1. - David Christopher, Aug 18 2014 The sequence is the INVERT transform of (1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). - Gary W. Adamson, Jul 16 2015 Also, number of threshold graphs on n nodes [Hougardy]. - Falk Hüffner, Dec 03 2015 Number of ternary words of length n in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017 a(n) is the number of words of length n over an alphabet of two letters, of which one letter appears an even number of times (the empty word of length 0 is included). See the analogous odd number case in A131577, and the Balakrishnan reference in A006516 (the 4-letter odd case), pp. 68-69, problems 2.66, 2.67 and 2.68. - Wolfdieter Lang, Jul 17 2017 Number of D-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are D-equivalent iff the positions of pattern D are identical in these paths. - Sergey Kirgizov, Apr 08 2018 Number of color patterns (set partitions) for an oriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if we permute the colors. For a(4)=8, the 4 achiral patterns are AAAA, AABB, ABAB, and ABBA; the 4 chiral patterns are the 2 pairs AAAB-ABBB and AABA-ABAA. - Robert A. Russell, Oct 30 2018 The determinant of the symmetric n X n matrix M defined by M(i,j) = (-1)^max(i,j) for 1 <= i,j <= n is equal to a(n) * (-1)^(n*(n+1)/2). - Bernard Schott, Dec 29 2018 For n>=1, a(n) is the number of permutations of length n whose cyclic representations can be written in such a way that when the cycle parentheses are removed what remains is 1 through n in natural order. For example, a(4)=8 since there are exactly 8 permutations of this form, namely, (1 2 3 4), (1)(2 3 4), (1 2)(3 4), (1 2 3)(4), (1)(2)(3 4), (1)(2 3)(4), (1 2)(3)(4), and (1)(2)(3)(4). Our result follows readily by conditioning on k, the number of parentheses pairs of the form ")(" in the cyclic representation. Since there are C(n-1,k) ways to insert these in the cyclic representation and since k runs from 0 to n-1, we obtain a(n) = Sum_{k=0..n-1} C(n-1,k) = 2^(n-1). - Dennis P. Walsh, May 23 2020 Maximum number of preimages that a permutation of length n + 1 can have under the consecutive-231-avoiding stack-sorting map. - Colin Defant, Aug 28 2020
REFERENCES
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7
Xavier Merlin, Methodix Algèbre, Ellipses, 1995, p. 153.
FORMULA
a(0) = 1, a(n) = 2^(n-1).
G.f.: (1 - x) / (1 - 2*x) = 1 / (1 - x / (1 - x)). - Michael Somos, Apr 18 2012 E.g.f.: cosh(z)*exp(z) = (exp(2*z) + 1)/2.
a(0) = 1 and for n>0, a(n) = sum of all previous terms.
a(n) = Sum_{k=0..n} binomial(n, 2*k). - Paul Barry, Feb 25 2003 a(n) = Sum_{k=0..n} binomial(n,k)*(1+(-1)^k)/2. - Paul Barry, May 27 2003 a(n) = floor((1+2^n)/2). - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003
G.f.: Sum_{i>=0} x^i/(1-x)^i. - Jon Perry, Jul 10 2004 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k+1, n-k)*binomial(2*k, k). - Paul Barry, Mar 18 2005 E.g.f.: (exp(2*x)+1)/2 = (G(0) + 1)/2; G(k) = 1 + 2*x/(2*k+1 - x*(2*k+1)/(x + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2011 A051049(n) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 18 2012
A008619(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 18 2012
G.f.: U(0), where U(k) = 1 + x*(k+3) - x*(k+2)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012 E.g.f.: E(0), where E(k) = 1 + x/( 2*k+1 - x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 25 2013 G.f.: 1 + x/(1 + x)*( 1 + 3*x/(1 + 3*x)*( 1 + 5*x/(1 + 5*x)*( 1 + 7*x/(1 + 7*x)*( 1 + ... )))). - Peter Bala, May 27 2017 a(n) = Sum_{k=0..2} stirling2(n, k).
a(n) = A053120(n, n), n >= 0, (main diagonal of triangle of Chebyshev's T polynomials). - Wolfdieter Lang, Nov 26 2019
EXAMPLE
G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 32*x^6 + 64*x^7 + 128*x^8 + ...
( -1 1 -1)
det ( 1 1 1) = 4
( -1 -1 -1)
MAPLE
with(PolynomialTools): A011782:=seq(coeftayl((1-x)/(1-2*x), x = 0, k), k=0..10^2); # Muniru A Asiru, Sep 26 2017
MATHEMATICA
f[s_] := Append[s, Ceiling[Plus @@ s]]; Nest[f, {1}, 32] (* Robert G. Wilson v, Jul 07 2006 *) CoefficientList[ Series[(1-x)/(1-2x), {x, 0, 32}], x] (* Robert G. Wilson v, Jul 07 2006 *) Table[Sum[StirlingS2[n, k], {k, 0, 2}], {n, 0, 30}] (* Robert A. Russell, Apr 25 2018 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 2^(n-1))};
(PARI) Vec((1-x)/(1-2*x) + O(x^30)) \\ Altug Alkan, Oct 31 2015 (Haskell)
a011782 n = a011782_list !! n
a011782_list = 1 : scanl1 (+) a011782_list
(Sage) [sum(stirling_number2(n, j) for j in (0..2)) for n in (0..35)] # G. C. Greubel, Jun 02 2020 (Python)
AUTHOR
Lee D. Killough (killough(AT)wagner.convex.com)
1, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152
COMMENTS
Powers of 2 doubled up. The usual OEIS policy is to omit the duplicates in such cases (when this would become A000079). This is an exception. Number of symmetric compositions of n: e.g., 5 = 2+1+2 = 1+3+1 = 1+1+1+1+1 so a(5) = 4; 6 = 3+3 = 2+2+2 = 1+4+1 = 2+1+1+2 = 1+2+2+1 = 1+1+2+1+1 = 1+1+1+1+1+1 so a(6) = 8. - Henry Bottomley, Dec 10 2001 Starting with offset 1 = binomial transform of [1, 1, -1, 3, -7, 17, -41, ...]; where A001333 = (1, 1, 3, 7, 17, 41, ...). - Gary W. Adamson, Mar 25 2009 a(n+1) is the number of symmetric subsets of [n]={1,2,...,n}. A subset S of [n] is symmetric if k is an element of S implies (n-k+1) is an element of S. - Dennis P. Walsh, Oct 27 2009 Pisano period lengths: 1, 1, 4, 1, 8, 4, 6, 1, 12, 8, 20, 4, 24, 6, 8, 1, 16, 12, 36, 8, ... . - R. J. Mathar, Aug 10 2012 Range of row n of the Circular Pascal array of order 4. - Shaun V. Ault, May 30 2014 a(n) is the number of permutations of length n avoiding both 213 and 312 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014 Also, the decimal representation of the diagonal from the origin to the corner (and from the corner to the origin except for the initial term) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 190", based on the 5-celled von Neumann neighborhood when initialized with a single black (ON) cell at stage zero. - Robert Price, May 10 2017 a(n + 1) + n - 1, n > 0, is the number of maximal subsemigroups of the monoid of partial order-preserving or -reversing mappings on a set with n elements. See the East et al. link. - James Mitchell and Wilf A. Wilson, Jul 21 2017 Number of symmetric stairs with n cells. A stair is a snake polyomino allowing only two directions for adjacent cells: east and north. See A005418. - Christian Barrientos, May 11 2018 For n >= 4, a(n) is the exponent of the group of the Gaussian integers in a reduced system modulo (1+i)^(n+2). See A302254. - Jianing Song, Jun 27 2018 a(n) is the number of length-(n+1) binary sequences, denoted <s(1),...,s(n+1)>, with s(1)=1 and with s(i+1)=s(i) for odd i. - Dennis P. Walsh, Sep 06 2018 a(n+1) is the number of subsets of {1,2,..,n} in which all differences between successive elements of subsets are even. For example, for n = 7, a(6) = 8 and the 8 subsets are {7}, {1,7}, {3,7}, {5,7}, {1,3,7}, {1,5,7}, {3,5,7}, {1,3,5,7}. For odd differences between elements see Comment in A000045 (Fibonacci numbers). - Enrique Navarrete, Jul 01 2020
LINKS
Emeric Deutsch, Problem 1633, Math. Mag., 74 #5 (2001), p. 403.
FORMULA
a(n) = a(n-1)*a(n-2)/a(n-3) = 2*a(n-2) = 2^ A004526(n). G.f.: (1+x)/(1-2*x^2).
a(n) = (1/2 + sqrt(1/8))*sqrt(2)^n + (1/2 - sqrt(1/8))*(-sqrt(2))^n. - Ralf Stephan, Mar 11 2003 E.g.f.: cosh(sqrt(2)*x) + sinh(sqrt(2)*x)/sqrt(2). - Paul Barry, Jul 16 2003 The signed sequence (-1)^n*2^floor(n/2) has a(n) = (sqrt(2))^n(1/2 - sqrt(2)/4) + (-sqrt(2))^n(1/2 + sqrt(2)/4). It is the inverse binomial transform of A000129(n-1). - Paul Barry, Apr 21 2004 Diagonal sums of A046854. a(n) = Sum_{k=0..n} binomial(floor(n/2), k). - Paul Barry, Jul 07 2004 a(n) = a(n-2) + 2^floor((n-2)/2). - Paul Barry, Jul 14 2004 a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), floor(k/2)). - Paul Barry, Jul 15 2004 E.g.f.: cosh(asinh(1) + sqrt(2)*x)/sqrt(2). - Michael Somos, Feb 28 2005 a(n) = 2^(n/2)*((1 + (-1)^n)/2 + (1-(-1)^n)/(2*sqrt(2))). - Paul Barry, Nov 12 2009 a(n) = 2^((2*n - 1 + (-1)^n)/4). - Luce ETIENNE, Sep 20 2014
EXAMPLE
For n=5 the a(5)=4 symmetric subsets of [4] are {1,4}, {2,3}, {1,2,3,4} and the empty set. - Dennis P. Walsh, Oct 27 2009 For n=5 the a(5)=4 length-6 binary sequences are <1,1,0,0,0,0>, <1,1,0,0,1,1>, <1,1,1,1,0,0> and <1,1,1,1,1,1>. - Dennis P. Walsh, Sep 06 2018
MATHEMATICA
With[{c=2^Range[0, 30]}, Riffle[c, c]] (* Harvey P. Dale, Jan 23 2015 *) CoefficientList[Series[(1+x)/(1-2*x^2), {x, 0, 50}], x] (* Stefano Spezia, Sep 07 2018 *)
PROG
(PARI) a(n)=if(n<0, 0, 2^(n\2))
(Maxima) makelist(2^floor(n/2), n, 0, 50); /* Martin Ettl, Oct 17 2012 */ (Sage)
x, y = -1, 0
while True:
yield -x
x, y = x + y, x - y
(Python)
CROSSREFS
See A052955 for partial sums (without the initial term). A000079 gives the odd-indexed terms of a(n).
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022
Rows of Losanitsch's triangle T(n, k), n >= 0, 0 <= k <= n.
+10
68
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 10, 9, 3, 1, 1, 4, 12, 19, 19, 12, 4, 1, 1, 4, 16, 28, 38, 28, 16, 4, 1, 1, 5, 20, 44, 66, 66, 44, 20, 5, 1, 1, 5, 25, 60, 110, 126, 110, 60, 25, 5, 1, 1, 6, 30, 85, 170, 236, 236, 170, 85, 30, 6, 1, 1, 6, 36, 110, 255
COMMENTS
Sometimes erroneously called "Lossnitsch's triangle". But the author's name is Losanitsch (I have seen the original paper in Chem. Ber.). This is a German version of the Serbian name Lozanic. - N. J. A. Sloane, Jun 29 2008 For n >= 3, a(n-3,k) is the number of series-reduced (or homeomorphically irreducible) trees which become a path P(k+1) on k+1 nodes, k >= 0, when all leaves are omitted (see illustration). Proof by Pólya's enumeration theorem. - Wolfdieter Lang, Jun 08 2001 The number of ways to put beads of two colors in a line, but take symmetry into consideration, so that 011 and 110 are considered the same. - Yong Kong (ykong(AT)nus.edu.sg), Jan 04 2005
Alternating row sums are 1,0,1,0,2,0,4,0,8,0,16,0,... - Gerald McGarvey, Oct 20 2008 The triangle sums, see A180662 for their definitions, link Losanitsch's triangle A034851 with several sequences, see the crossrefs. We observe that the Ze3 and Ze4 sums link Losanitsch's triangle with A005683, i.e., R. K. Guy's Twopins game. - Johannes W. Meijer, Jul 14 2011 T(n-(L-1)k, k) is the number of ways to cover an n-length line by exactly k L-length segments excluding symmetric covers. For L=2 it is corresponds to A102541, for L=3 to A228570 and for L=4 to A228572. - Philipp O. Tsvetkov, Nov 08 2013 Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 1 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014 T(n, k) is the number of non-isomorphic outer planar graphs of order n+3, size n+3+k, and maximum degree k+2. - Christian Barrientos, Oct 18 2018 T(n, k) is the sum of even-degree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, n-k) is even for T(n, k) paths and odd for A034852(n, k) of them. For a (non-reversible) string of k black and n-k white beads, consider the minimum number of bead transpositions needed to place the black ones to the left and the white ones to the right (in other words, the number of inversions of the permutation obtained by labeling the black beads by integers 1,...,k and the white ones by k+1,...,n, in the same order they take on the string). It is even for T(n, k) strings and odd for A034852(n, k) cases. (End)
Named after the Serbian chemist, politician and diplomat Simeon Milivoje "Sima" Lozanić (1847-1935). - Amiram Eldar, Jun 10 2021 T(n, k) is the number of caterpillars with a perfect matching, with 2n+2 vertices and diameter 2n-1-k. - Christian Barrientos, Sep 12 2023
LINKS
Tewodros Amdeberhan, Mahir Bilen Can and Victor H. Moll, Broken bracelets, Molien series, paraffin wax and the elliptic curve of conductor 48, SIAM Journal of Discrete Math., Vol. 25, No. 4 (2011), p. 1843-1859; arXiv preprint, arXiv:1106.4693 [math.CO], 2011. See Theorem 2.8. Stephen G. Hartke and A. J. Radcliffe, Signatures of Strings, Annals of Combinatorics, Vol. 17, No. 1 (March, 2013), pp. 131-150.
FORMULA
G.f. for k-th column (if formatted as lower triangular matrix a(n, k)): x^k*Pe(floor((k+1)/2), x^2)/(((1-x)^(k+1))*(1+x)^(floor((k+1)/2))), where Pe(n, x^2) := Sum_{m=0..floor(n/2)} A034839(n, m)*x^(2*m) (row polynomials of Pascal array even numbered columns). - Wolfdieter Lang, May 08 2001 a(n, k) = a(n-1, k-1) + a(n-1, k) - C(n/2-1, (k-1)/2), where the last term is present only if n is even and k is odd (see Sloane link).
T(n, k) = T(n-2, k-2) + T(n-2, k) + C(n-2, k-1), n > 1.
Let P(n, x, y) = Sum_{m=0..n} a(n, m)*x^m*y^(n-m), then for x > 0, y > 0 we have P(n, x, y) = (x+y)*P(n-1, x, y) for n odd and P(n, x, y) = (x+y)*P(n-1, x, y) - x*y*(x^2+y^2)^((n-2)/2) for n even. - Gerald McGarvey, Feb 15 2005 It appears that:
T(n,k) = C(n,k)/2, n even, k odd;
T(n,k) = (C(n,k) + C(n/2,k/2))/2, n even, k even;
T(n,k) = (C(n,k) + C((n-1)/2,(k-1)/2))/2, n odd, k odd;
T(n,k) = (C(n,k) + C((n-1)/2,k/2))/2, n odd, k even.
(End)
EXAMPLE
Triangle begins
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 2, 4, 2, 1;
1, 3, 6, 6, 3, 1;
1, 3, 9, 10, 9, 3, 1;
1, 4, 12, 19, 19, 12, 4, 1;
1, 4, 16, 28, 38, 28, 16, 4, 1;
1, 5, 20, 44, 66, 66, 44, 20, 5, 1;
MAPLE
A034851 := proc(n, k) option remember; local t; if k = 0 or k = n then return(1) fi; if n mod 2 = 0 and k mod 2 = 1 then t := binomial(n/2-1, (k-1)/2) else t := 0; fi; A034851(n-1, k-1)+ A034851(n-1, k)-t; end: seq(seq( A034851(n, k), k=0..n), n=0..11);
MATHEMATICA
t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_, k_] := (Binomial[n, k] + Binomial[Quotient[n, 2], Quotient[k, 2]])/2; Flatten[Table[t[n, k], {n, 0, 12}, {k, 0, n}]](* Jean-François Alcover, Feb 07 2012, after PARI *)
PROG
(PARI) {T(n, k) = (1/2) *(binomial(n, k) + binomial(n%2, k%2) * binomial(n\2, k\2))}; /* Michael Somos, Oct 20 1999 */ (Haskell)
a034851 n k = a034851_row n !! k
a034851_row 0 = [1]
a034851_row 1 = [1, 1]
a034851_row n = zipWith (-) (zipWith (+) ([0] ++ losa) (losa ++ [0]))
([0] ++ a204293_row (n-2) ++ [0])
where losa = a034851_row (n-1)
a034851_tabl = map a034851_row [0..]
CROSSREFS
Triangle sums (see the comments): A005418 (Row), A011782 (Related to Row2), A102526 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23), A005207 (Kn3, Kn4), A005418 (Fi1, Fi2), A102543 (Ca1, Ca2), A192928 (Gi1, Gi2), A005683 (Ze3, Ze4). Sums of squares of terms in rows equal A211208.
a(-1) = 1; for n >= 0, a(n) = 2^n + 4^n = 2^n*(1 + 2^n).
+10
61
1, 2, 6, 20, 72, 272, 1056, 4160, 16512, 65792, 262656, 1049600, 4196352, 16781312, 67117056, 268451840, 1073774592, 4295032832, 17180000256, 68719738880, 274878431232, 1099512676352, 4398048608256, 17592190238720, 70368752566272
COMMENTS
Counts closed walks of length 2n+2 at a vertex of the cyclic graph on 8 nodes C_8.
The count of closed walks of odd length is zero. See the array w(N,L) and triangle a(K,N) given in A199571 for the general case. - Wolfdieter Lang, Nov 08 2011 Number of monic irreducible polynomials of degree 1 in GF(2^n)[x,y]. - Max Alekseyev, Jan 23 2006 a(n) written in base 2: a(-1) = 1, a(0) = 10, a(n) for n >= 1: 110, 10100, 1001000, 100010000, ..., i.e., number 1, (n-1) times 0, number 1, n times 0 (see A163664). a(n) for n >= 0 is duplicate of A161168. a(n) for n >= 0 is a bisection of A005418. - Jaroslav Krizek, Aug 14 2009
REFERENCES
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121. See Table 4.
FORMULA
a(n) = Sum_{k=0..n} if((n-k) mod 4 = 0, binomial(n, 2*k), 0)}. - Paul Barry, Sep 19 2005 G.f.: 1/x + (2-6*x)/((1-2*x)*(1-4*x)).
E.g.f.: (1+Q(0))/4, where Q(k) = 1 + 2/( 2^k - 2*x*4^k/( 2*x*2^k + (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013
EXAMPLE
a(1)=6 counts six round trips from, say, vertex no 1: 12121, 18181, 12181, 18121, 12321 and 18781. - Wolfdieter Lang, Nov 08 2011
MATHEMATICA
Table[2^n + 4^n, {n, 0, 25}]
PROG
(PARI) a(n)={if(n>=0, 2^n*(1 + 2^n), 1)} \\ Harry J. Smith, Aug 20 2009 (PARI) {a(n)=n!*polcoeff((1 + exp(2*x +x*O(x^n)))^2/4, n)} \\ Paul D. Hanna, Oct 25 2012
CROSSREFS
Cf. A000051, A006516, A007582, A034472, A034474, A034491, A052539, A062394, A062395, A062396, A007689, A063376, A063481, A074600 - A074624, A122983.
Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (2,n)-rectangular grid with k '1's and (2n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
+10
41
1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 6, 6, 6, 2, 1, 1, 2, 10, 14, 22, 14, 10, 2, 1, 1, 3, 15, 32, 60, 66, 60, 32, 15, 3, 1, 1, 3, 21, 55, 135, 198, 246, 198, 135, 55, 21, 3, 1, 1, 4, 28, 94, 266, 508, 777, 868, 777, 508, 266, 94, 28, 4, 1, 1, 4, 36
COMMENTS
Sum of rows (see example) gives A225826. By columns:
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014
EXAMPLE
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1
1 1 1 1
2 1 1 3 1 1
3 1 2 6 6 6 2 1
4 1 2 10 14 22 14 10 2 1
5 1 3 15 32 60 66 60 32 15 3 1
6 1 3 21 55 135 198 246 198 135 55 21 3 1
7 1 4 28 94 266 508 777 868 777 508 266 94 28 4 1
8 1 4 36 140...
...
The length of row n is 2*n+1, so n>= floor((k+1)/2).
MAPLE
if type(k, 'even') then
binomial(2*n, k) +3*binomial(n, k/2) ;
else
binomial(2*n, k) +(1-(-1)^n)*binomial(n-1, (k-1)/2) ;
end if ;
%/4 ;
end proc:
MATHEMATICA
T[n_, k_] := If[EvenQ[k],
Binomial[2n, k] + 3 Binomial[n, k/2],
Binomial[2n, k] + (1-(-1)^n) Binomial[n-1, (k-1)/2]]/4;
1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152
COMMENTS
a(n) is also the number of median-reflective (palindrome) symmetric patterns in a top-down equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.
The number of possibilities for an n-game (sub)set of tennis with neither player gaining a 2-game advantage. (Motivated by the marathon Isner-Mahut match at Wimbledon, 2010.) - Barry Cipra, Jun 28 2010 Number of achiral rows of n colors using up to two colors. For a(3)=4, the rows are AAA, ABA, BAB, and BBB. - Robert A. Russell, Nov 07 2018
FORMULA
a(n) = 2^ceiling(n/2).
E.g.f.: cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Feb 02 2023
MAPLE
for n from 0 to 100 do printf(`%d, `, 2^ceil(n/2)) od:
MATHEMATICA
2^Ceiling[Range[0, 50]/2] (* or *) Riffle[2^Range[0, 25], 2^Range[25]] (* Harvey P. Dale, Mar 05 2013 *)
PROG
(PARI) { for (n=0, 500, write("b060546.txt", n, " ", 2^ceil(n/2)); ) } \\ Harry J. Smith, Jul 06 2009 (Magma) [2^Ceiling(n/2): n in [0..50]]; // G. C. Greubel, Nov 07 2018
CROSSREFS
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022
AUTHOR
André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (3,n)-rectangular grid with k '1's and (3n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
+10
35
1, 1, 2, 2, 1, 1, 2, 6, 6, 6, 2, 1, 1, 4, 13, 27, 39, 39, 27, 13, 4, 1, 1, 4, 22, 60, 139, 208, 252, 208, 139, 60, 22, 4, 1, 1, 6, 34, 129, 371, 794, 1310, 1675, 1675, 1310, 794, 371, 129, 34, 6, 1, 1, 6, 48, 218, 813, 2196, 4767, 8070, 11139, 12300, 11139, 8070, 4767, 2196, 813, 218, 48, 6, 1
COMMENTS
Sum of rows (see example) gives A225827. By columns:
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 3 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015
EXAMPLE
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1
1 1 2 2 1
2 1 2 6 6 6 2 1
3 1 4 13 27 39 39 27 13 4 1
4 1 4 22 60 139 208 252 208 139 60 22 4 1
5 1 6 34 129 371 794 1310 1675 1675 1310 794 371 129 34 6 1
6 1 6 48 218 813 2196 4767 8070 11139 12300 11139 8070 4767 2196 813 218 48 6 1
...
The length of row n is 3*n+1.
MATHEMATICA
T[n_, k_] := (Binomial[3n, k] + If[OddQ[n] || EvenQ[k], Binomial[Quotient[3 n, 2], Quotient[k, 2]], 0] + Sum[Binomial[n, k - 2i] Binomial[n, i] + Binomial[3 Mod[n, 2], k - 2i] Binomial[3 Quotient[n, 2], i], {i, 0, Quotient[k, 2]}])/4; Table[T[n, k], {n, 0, 6}, {k, 0, 3n}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
PROG
(PARI)
T(n, k)={(binomial(3*n, k) + if(n%2==1||k%2==0, binomial(3*n\2, k\2), 0) + sum(i=0, k\2, binomial(n, k-2*i) * binomial(1*n, i) + binomial(3*(n%2), k-2*i) * binomial(3*(n\2), i)))/4}
for(n=0, 6, for(k=0, 3*n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 30 2017
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