Displaying 21-30 of 88 results found.
Singular n X n (0,1)-matrices: the number of n X n (0,1)-matrices having distinct, nonzero ordered rows, but having at least two equal columns or at least one zero column.
(Formerly M4306 N1801)
+10
7
0, 6, 350, 43260, 14591171, 14657461469, 46173502811223, 474928141312623525, 16489412944755088235117, 1985178211854071817861662307, 846428472480689964807653763864449, 1299141117072945982773752362381072143359, 7268140170419155675761326840423792818571154945, 149650282980396792665043455999899697765782372693740287
COMMENTS
This is a lower bound for the set of all n X n (0,1)-matrices having distinct, nonzero ordered rows and determinant 0 (compare A000410). Here ordered means that we take only one representative from the n! matrices obtained by all permutations of the distinct rows of an n X n matrix.
a(n) is also the number of sets of n distinct nonzero (0,1)-vectors in R^n that do not span R^n.
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = (-1)*Sum_{k=0..n-1} Stirling1(n+1, k+1)*binomial(2^k-1, n).
MAPLE
with(combinat): T := proc(n) -sum(stirling1(n+1, k+1)*binomial(2^k-1, n), k=0..n-1); end proc:
MATHEMATICA
a[n_] := -Sum[ StirlingS1[n+1, k+1]*Binomial[2^k-1, n], {k, 0, n-1}]; Table[a[n], {n, 2, 15}] (* Jean-François Alcover, Nov 21 2012, from formula *)
PROG
(PARI) a(n) = -sum(k=0, n-1, stirling(n+1, k+1, 1)*binomial(2^k-1, n)); \\ Michel Marcus, Jun 05 2020 (Magma) [ -(&+[StirlingFirst(n+1, k+1)*Binomial(2^k-1, n): k in [0..n-1]]): n in [2..15]]; // G. C. Greubel, Jun 05 2020 (Sage) [sum((-1)^(n+k+1)*stirling_number1(n+1, k+1)*binomial(2^k-1, n) for k in (0..n-1)) for n in (2..15)] # G. C. Greubel, Jun 05 2020
Nonsingular n X n matrices over GF(8).
+10
7
1, 7, 3528, 115379712, 241909719367680, 32467582052437076213760, 278893342293098904613804037898240, 153323163270070838469523866093442017326530560
FORMULA
a(n) = (8^n - 1)*(8^n - 8)*...*(8^n - 8^(n-1)).
MATHEMATICA
Table[Product[(8^n - 8^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)
PROG
(Magma) [1] cat [&*[(8^n-8^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 30 2013 (PARI) {a(n) = prod(j=0, n-1, 8^n - 8^j)}; \\ G. C. Greubel, May 14 2019 (Sage) [product(8^n - 8^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019
Nonsingular n X n matrices over GF(9).
+10
7
1, 8, 5760, 339655680, 1624314979123200, 629282246371356907929600, 19747506525777609095698646040576000, 50195501537943419769100848121708339934527488000
FORMULA
a(n) = (9^n - 1)*(9^n - 9)*...*(9^n - 9^(n-1)).
MATHEMATICA
Table[Product[(9^n - 9^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)
PROG
(Magma) [1] cat [&*[(9^n - 9^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013 (PARI) {a(n) = prod(j=0, n-1, 9^n - 9^j)}; \\ G. C. Greubel, May 14 2019 (Sage) [product(9^n - 9^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019
Number of nonsingular n X n (-1,0,1)-matrices (over the reals).
+10
7
1, 2, 48, 11808, 27947520, 609653621760, 119288919620689920
COMMENTS
It would be nice to have an estimate for the asymptotic rate of growth.
EXAMPLE
a(1) = 2: [1], [ -1].
a(2) = 48: There are 8 choices for the first column, u (say) and then the 2nd column can be anything except 0, u, -u, so 6 choices, giving a total of 8*6 = 48.
MATHEMATICA
(* A brute force solution up to n = 4 *) a[n_] := a[n] = (m = Array[x, {n, n}]; cnt = 0; iter = {#, -1, 1}& /@ Flatten[m]; Do[ If[ Det[m] != 0, cnt++], Evaluate[ Sequence @@ iter]]; cnt); Table[ Print[a[n]]; a[n], {n, 1, 4}] (* Jean-François Alcover, Oct 11 2012 *)
EXTENSIONS
a(4) from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000
a(0)-a(5) confirmed and a(6) added by Minfeng Wang, May 01 2024
Vandermonde determinant of the first n terms of (1,2,4,8,16,...).
+10
6
1, 1, 6, 1008, 20321280, 203199794380800, 4096245678214226116608000, 671169825411994707343327912777482240000, 3589459026274030507466469204160461571257625328222208000000, 2511229721141086754031154605327661795863172723306019839389105937236728217600000000
COMMENTS
Each term divides its successor, as in A002884. Indeed, 2*v(n+1)/v(n) divides v(n+2)/v(n+1), as in A171499.
FORMULA
a(n) = Product_{0 <= i < j <= n-1} (2^j - 2^i).
a(n) = 2^(n*(n-1)*(n-2)/6) * Product_{1<=k<=n-1} (2^k-1)^(n-k). (End)
a(n) = Product_{k=0..n-2} ( 2^(k+1)^2 * QPochhammer(2^(-k-1); 2; k+1) ). - G. C. Greubel, Aug 31 2023
MAPLE
# First program
with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([2^i$i=0..n-1])):
# Second program
f:= n -> 2^(n*(n-1)*(n-2)/6)*mul((2^k-1)^(n-k), k=1..n-1):
MATHEMATICA
(* First program *)
f[j_]:= 2^(j-1); z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j, k-1}], {k, 2, n}]
Table[v[n+1]/v[n], {n, z}] (* A002884 *) Table[v[n]*v[n+2]/(2*v[n+1]^2), {n, z}] (* A171499 *) Table[FactorInteger[v[n]], {n, z}]
(* Second program *)
Table[Product[2^(k+1) -2^j, {k, 0, n-2}, {j, 0, k}], {n, 15}] (* G. C. Greubel, Aug 31 2023 *)
PROG
(Magma) [1] cat [(&*[(&*[2^(k+1) -2^j: j in [0..k]]): k in [0..n-2]]): n in [2..15]]; // G. C. Greubel, Aug 31 2023 (SageMath) [product(product(2^(k+1) -2^j for j in range(k+1)) for k in range(n-1)) for n in range(1, 16)] # G. C. Greubel, Aug 31 2023
Number of labeled groups.
+10
5
1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, 259459200, 518918400, 16605388800, 163459296000, 10353459916800, 22230464256000, 1867358997504000, 6758061133824000, 648773868847104000, 5474029518397440000, 122618261212102656000
COMMENTS
In other words, number of ways to define a group structure on a set of n elements. Note that for a group G, a group structure on the set G is given by mapping (x,y) to sigma^(-1)(sigma(x)*sigma(y)), where sigma is a bijection on the set G; sigma and sigma' give the same structure if and only if sigma' is the composition of a group automorphism of G and sigma.
By definition, a(n) = A034381(n) if n in A003277, otherwise a(n) > A034381(n). The indices of records of a(n)/ A034381(n) among the known terms are 1, 4, 8, 16, 24, 32, 48, 64, 96, 128, 192, with a(192)/ A034381(192) = 122774329/1640520 ~ 74.8. Also by definition, a(n) >= A000001(n)*n!/ A059773(n). If the conjecture A059773(2^r) = A002884(r) is true, then A059773(2^r) <= 2^(r^2), while A000001(2^r) >= 2^((2/27)*r^2*(r-6)) (see the Math Stack Exchange link below), so a(2^r)/ A034381(2^r) tends to infinity quickly as r tends to infinity. The sequence is strictly increasing for the first 256 terms (a(256) > A034381(256) > A034381(255) = a(255) since 255 is in A003277). On the other hand, assuming that A059773(2^r) = A002884(r), then a(2^20)/(2^20)! >= A000001(2^20)/ A002884(20) > 99798.4, while a(2^20+1)/(2^20)! = A034381(2^20+1)/(2^20)! = (2^20+1)/phi(2^20+1) since 2^20+1 = 17*61681 is in A003277, so we would have a(2^20) > a(2^20+1). It is conjectured a(2^r) > a(2^r+1) for all sufficiently large r. (End)
FORMULA
a(n) = Sum n!/|Aut(G)|, where the sum is taken over the different products G of cyclic groups with |G| = n.
PROG
(GAP) A034383 := function(n) local fn, sum, k; fn := Factorial(n); sum := 0; for k in [1 .. NrSmallGroups(n)] do sum := sum + fn / Size(AutomorphismGroup(SmallGroup(n, k))); od; return sum; end; # Stephen A. Silver, Feb 10 2013
Number of nonsingular n X n matrices over GF(11).
+10
5
1, 10, 13200, 2124276000, 41393302251840000, 97602635428252959312000000, 27847155251069188894843979022720000000, 961359275427083998992553051820498439890246400000000
FORMULA
a(n) = (11^n - 1)*(11^n - 11)*...*(11^n - 11^(n-1)).
MATHEMATICA
Table[Product[11^n - 11^k, {k, 0, n-1}], {n, 0, 10}] (* Vincenzo Librandi, Jan 28 2013 *)
PROG
(Magma) [1] cat [&*[(11^n - 11^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013 (PARI) {a(n) = prod(j=0, n-1, 11^n - 11^j)}; \\ G. C. Greubel, May 14 2019 (Sage) [product(11^n - 11^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019
Let A_n be the upper triangular matrix in the group GL(n,2) that has zero entries below the main diagonal and 1 elsewhere; a(n) is the size of the conjugacy class of this matrix in GL(n,2).
+10
5
1, 3, 42, 2520, 624960, 629959680, 2560156139520, 41781748196966400, 2732860586067178291200, 715703393163961188325785600, 750102961052993818881476159078400, 3145391744524297920839316348340273152000, 52764474940208177704130232748554603290689536000
EXAMPLE
For example for n=4 the matrix is / 1,1,1,1 / 0,1,1,1 / 0,0,1,1 / 0,0,0,1 /.
MAPLE
a:= n-> 2^((n-1)*(n-2)/2) *mul(2^k-1, k=1..n):
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Jun 12 2003
Triangle read by rows: T(n,k) is the number of n X n matrices of rank k over F_2.
+10
5
1, 1, 1, 1, 9, 6, 1, 49, 294, 168, 1, 225, 7350, 37800, 20160, 1, 961, 144150, 4036200, 19373760, 9999360, 1, 3969, 2542806, 326932200, 8543828160, 39687459840, 20158709760, 1, 16129, 42677334, 23435953128, 2812314375360, 71124337751040, 325139829719040, 163849992929280
FORMULA
T(n,k) = Product_{j=0..k-1} (2^n - 2^j)^2/(2^k - 2^j) = A022166(n,k) * Product_{j=0..k-1} (2^n - 2^j).
EXAMPLE
Triangle T(n,k) begins:
1;
1, 1;
1, 9, 6;
1, 49, 294, 168;
1, 225, 7350, 37800, 20160;
1, 961, 144150, 4036200, 19373760, 9999360;
...
T(2,1) = 9 because there are 9, 2 X 2 matrices in F_2 that have rank 1: {{0, 0}, {0, 1}}, {{0, 0}, {1, 0}}, {{0, 0}, {1, 1}}, {{0, 1}, {0, 0}}, {{0, 1}, {0, 1}}, {{1, 0}, {0, 0}}, {{1, 0}, {1, 0}}, {{1,1}, {0, 0}}, {{1, 1}, {1, 1}}.
MAPLE
T:= (n, k) -> mul((2^n-2^j)^2/(2^k-2^j), j=0..k-1):
MATHEMATICA
q = 2; Table[Table[Product[(q^n - q^i)^2/(q^k - q^i), {i, 0, k - 1}], {k, 0, n}], {n, 0, 6}] // Grid
Triangle read by rows: T(n,k) is the number of diagonalizable n X n matrices over GF(2) that have rank k, n >= 0, 0 <= k <= n.
+10
5
1, 1, 1, 1, 6, 1, 1, 28, 28, 1, 1, 120, 560, 120, 1, 1, 496, 9920, 9920, 496, 1, 1, 2016, 166656, 714240, 166656, 2016, 1, 1, 8128, 2731008, 48377856, 48377856, 2731008, 8128, 1, 1, 32640, 44216320, 3183575040, 13158776832, 3183575040, 44216320, 32640, 1
COMMENTS
Equivalently, T(n,k) is the number of n X n matrices, P, over GF(2) with rank k, such that P^2 = P.
Equivalently, T(n,k) is the number of direct sum decompositions of the vector space GF(2)^n into exactly two subspaces U and W such that the dimension of U is k.
FORMULA
T(n,k)/ A002884(n) is the coefficient of y^k*x^n in the expansion of Sum_{n>=0} x^n\ A002884(n) * Sum_{n>=0} y*x^n\ A002884(n).
EXAMPLE
Triangle begins:
1;
1, 1;
1, 6, 1;
1, 28, 28, 1;
1, 120, 560, 120, 1;
1, 496, 9920, 9920, 496, 1;
1, 2016, 166656, 714240, 166656, 2016, 1;
MATHEMATICA
nn = 8; g[n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,
Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}] Sum[z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]]
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