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Mapping from the column-by-column reading to the half-antidiagonal reading of the triangular tables. Inverse of A056536.
+20
5
1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 11, 15, 20, 25, 10, 14, 19, 24, 30, 36, 13, 18, 23, 29, 35, 42, 49, 17, 22, 28, 34, 41, 48, 56, 64, 21, 27, 33, 40, 47, 55, 63, 72, 81, 26, 32, 39, 46, 54, 62, 71, 80, 90, 100, 31, 38, 45, 53, 61, 70, 79, 89, 99, 110, 121, 37, 44, 52, 60, 69
COMMENTS
As a square array, this is the dispersion of the complement of the squares; see A082152. - Clark Kimberling, Apr 05 2003
FORMULA
Triangle T(n, k), 1<=k<=n, read by rows, defined by: T(n, k) = 0 for n<k and T(n, k) = A002620(n-k+1) + k*n + k - n if n>=k. T(n, n) = n^2; T(n, 1) = 1 + A002620(n) = A033638(n). - Philippe Deléham, Feb 16 2004 Square: t(n,k) = (n-1)(n+k) + k^2/4 + (1/8)(7+(-1)^k). - Clark Kimberling, Aug 08 2013
EXAMPLE
As a square array, a northwest corner:
1 ... 2 ... 3 ... 5 ... 7 ... 10
4 ... 6 ... 8 ... 11 .. 14 .. 18
9 ... 12 .. 15 .. 19 .. 23 .. 28
16 .. 20 .. 24 .. 29 .. 34 .. 40
25 .. 30 .. 35 .. 41 .. 47 .. 54
36 .. 42 .. 48 .. 55 .. 62 .. 70
49 .. 56 .. 63 .. 71 .. 79 .. 88
64 .. 72 .. 80 .. 89 .. 98 .. 108
MAPLE
# using Maple procedure nthmember given in A054426: [seq(nthmember(j, A056536), j=1..105)];
MATHEMATICA
(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := n+Floor[1/2+Sqrt[n]] (* complement of column 1 *); mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]; rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]] (* A056537 array *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A056537 sequence *)
CROSSREFS
Cf. A185787 (dispersion of complement of triangular numbers). Cf. A082152 (dispersion of complement of pentagonal numbers).
Dispersion of the complement of row 1 of A056536.
+20
3
1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 11, 7, 36, 30, 24, 19, 14, 10, 49, 42, 35, 29, 23, 18, 13, 64, 56, 48, 41, 34, 28, 22, 17, 81, 72, 63, 55, 47, 40, 33, 27, 21, 100, 90, 80, 71, 62, 54, 46, 39, 32, 26, 121, 110, 99, 89, 79, 70, 61, 53, 45, 38, 31, 144, 132, 120, 109
COMMENTS
Rectangular array read by antidiagonals; a permutation of the natural numbers. (Row 1) = squares = A000290(n) = n^2. (Dispersion of complement of column 1 of A082156) = (Transpose of A056537). The associated fractal sequence is A122196.
EXAMPLE
Northwest corner:
1 4 9 16 25
2 6 12 20 30
3 8 15 24 35
5 11 19 29 41
7 14 23 34 47
Catalan triangle read by rows. Also triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).
+10
32
1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 9, 5, 1, 6, 14, 14, 1, 7, 20, 28, 14, 1, 8, 27, 48, 42, 1, 9, 35, 75, 90, 42, 1, 10, 44, 110, 165, 132, 1, 11, 54, 154, 275, 297, 132, 1, 12, 65, 208, 429, 572, 429, 1, 13, 77, 273, 637, 1001, 1001, 429, 1, 14, 90, 350, 910, 1638, 2002, 1430, 1, 15, 104
COMMENTS
Number of standard tableaux of shape (n-k,k) (0<=k<=floor(n/2)). Example: T(4,1)=3 because in th top row we can have 124, 134, or 123 (but not 234). - Emeric Deutsch, May 23 2004 T(n,k) is the number of n-digit binary words (length n sequences on {0,1}) containing k 1's such that no initial segment of the sequence has more 1's than 0's. - Geoffrey Critzer, Jul 31 2009 T(n,k) is the number of dispersed Dyck paths (i.e. Motzkin paths with no (1,0) steps at positive heights) of length n and having k (1,1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HHHUD, HHUDH, HUDHH, and UDHHH. - Emeric Deutsch, May 30 2011 T(n,k) is the number of length n left factors of Dyck paths having k (1,-1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), we have UUUUD, UUUDU, UUDUU, and UDUUU. There is a simple bijection between length n left factors of Dyck paths and dispersed Dyck paths of length n, that takes D steps into D steps. - Emeric Deutsch, Jun 19 2011 Triangle, with zeros omitted, given by (1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...) DELTA (0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Chassidy Bozeman, Christine Cheng, Pamela E. Harris, Stephen Lasinis, and Shanise Walker, The Pinnacle Sets of a Graph, arXiv:2406.19562 [math.CO], 2024. See pp. 9-10. L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90. [Annotated scanned copy]
FORMULA
T(n, 0) = 1 if n >= 0; T(2*k, k) = T(2*k-1, k-1) if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) if k=1, 2, ..., floor(n/2). - Michael Somos, Aug 17 1999 T(n, k) = binomial(n, k) - binomial(n, k-1). - Michael Somos, Aug 17 1999 Sum_{k=0..n} T(n,k)*x^k = A000012(n), A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 12 2011
EXAMPLE
Triangle begins:
1;
1;
1, 1;
1, 2;
1, 3, 2;
1, 4, 5;
1, 5, 9, 5;
1, 6, 14, 14;
1, 7, 20, 28, 14;
...
T(5,2) = 5 because there are 5 such sequences: {0, 0, 0, 1, 1}, {0, 0, 1, 0, 1}, {0, 0, 1, 1, 0}, {0, 1, 0, 0, 1}, {0, 1, 0, 1, 0}. - Geoffrey Critzer, Jul 31 2009
MAPLE
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
end:
T:= (n, k)-> b(n, n-2*k):
MATHEMATICA
Table[Binomial[k, i]*(k - 2 i + 1)/(k - i + 1), {k, 0, 20}, {i, 0, Floor[k/2]}] // Grid (* Geoffrey Critzer, Jul 31 2009 *)
PROG
(PARI) {T(n, k) = if( k<0 || k>n\2, 0, if( n==0, 1, T(n-1, k-1) + T(n-1, k)))}; /* Michael Somos, Aug 17 1999 */ (Haskell)
a008315 n k = a008315_tabf !! n !! k
a008315_row n = a008315_tabf !! n
a008315_tabf = map reverse a008313_tabf
CROSSREFS
T(2n, n) = A000108 (Catalan numbers), row sums = A001405 (central binomial coefficients).
Consider the triangle shown below; sequence contains the concatenation of numbers read at a 45-degree angle upwards with horizontal beginning with the first term of a row.
+10
2
1, 2, 43, 75, 1186, 16129, 22171310, 29231814, 3730241915, 4638312520, 564739322621, 675748403327, 79685849413428, 92806959504235, 10693817060514336, 121107948271615244, 137122108958372625345, 1541381231099684736354, 1721551391241109785746455
COMMENTS
1
2 3
4 5 6
7 8 9 10
11 12 13 14 15
16 17 18 19 20 21
...
a(n) also is the concatenation of the terms of the n-th row of A056536. - Michel Marcus, Dec 14 2023
MAPLE
read("transforms"):
A079823aux := proc(n, k)
end proc:
local L, k, n0 ;
n0 := n-1 ;
L := [] ;
for k from 0 do
if k > n0-k then
break;
end if;
L := [op(L), A079823aux(n0-k, k)] ;
end do:
digcatL(L) ;
# second Maple program:
T:= (i, j)-> i*(i-1)/2+j:
a:= n-> parse(cat(seq(T(n-j, j+1), j=0..(n-1)/2))):
MATHEMATICA
Table[FromDigits[Join@@IntegerDigits[Table[Binomial[n-k+1, 2] + k, {k, Ceiling[n/2]}]]], {n, 30}] (* G. C. Greubel, Dec 13 2023 *)
EXTENSIONS
More terms from Jason D. W. Taff (jtaff(AT)jburroughs.org), Oct 31 2003
Numbers in n-th upward diagonal of triangle T : 0; 1, 2; 3, 4, 5; 6, 7, 8, 9; ...
+10
2
0, 1, 3, 2, 6, 4, 10, 7, 5, 15, 11, 8, 21, 16, 12, 9, 28, 22, 17, 13, 36, 29, 23, 18, 14, 45, 37, 30, 24, 19, 55, 46, 38, 31, 25, 20, 66, 56, 47, 39, 32, 26, 78, 67, 57, 48, 40, 33, 27, 91, 79, 68, 58, 49, 41, 34, 105, 92, 80, 69, 59, 50, 42, 35, 120, 106, 93, 81, 70, 60, 51
COMMENTS
See also A090894 (downward diagonal).
EXAMPLE
0; 1; 3,2; 6,4; 10,7,5; 15,11,8; 21,16,12,9; 28,22,17,13; ...
Array T(n,k) = n*(n+k), read by antidiagonals.
+10
1
0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0
EXAMPLE
Table begins
0;
1, 0;
4, 2, 0;
9, 6, 3, 0;
16, 12, 8, 4, 0;
25, 20, 15, 10, 5, 0;
36, 30, 24, 18, 12, 6, 0;
...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.
MAPLE
seq(seq((j-i)*j, i=0..j), j=0..14);
MATHEMATICA
Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)
PROG
(GAP) Flat(List([0..11], j->List([0..j], i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018
CROSSREFS
Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.
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