A132202 - OEIS

%I #19 Oct 21 2023 04:14:15

%S 1,1860,90291600,31082452632000,46764764308702440000,

%T 229747284991066934931840000,3031982831164890119435183865600000,

%U 93453554057243260025029337978773248000000,6055976192395031960092036887782708145734400000000,760152286561053082358524425840024164536832608896000000000

%N Number of 3n X 2n (0,1)-matrices with every row sum 2 and column sum 3.

%D Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.

%H G. C. Greubel, <a href="/A132202/b132202.txt">Table of n, a(n) for n = 1..85</a>

%F a(n) = f(3*n, 2*n), where f(m, n) = 2^(-m) * Sum_{j=0..n} (-1)^j*n!*m!*(2*m-2*j)!/(j!*(m-j)!*(n-j)!*6^(n-j)).

%F From _G. C. Greubel_, Oct 12 2023: (Start)

%F a(n) = ((6*n)!/(288)^n)*Sum_{j=0..2*n} b(2*n,j)*b(3*n,j)*(-6)^j/(j!*b(2*j, j)*b(6*n,2*j)), where b(x,y) = binomomial(x,y).

%F a(n) = (6*n)!/(288)^n * Hypergeometric1F1([-2*n], [1/2-3*n], -3/2). (End)

%F a(n) ~ sqrt(Pi) * 2^(n+1) * 3^(4*n + 1/2) * n^(6*n + 1/2) / exp(6*n+1). - _Vaclav Kotesovec_, Oct 21 2023

%e 1 for 3X2:

%e   11

%e   11

%e   11

%e 1860 for 6X4.

%e 90291600 for 9X6.

%p f:=proc(m,n) 2^(-m)*add( ((-1)^(i)*m!*n!*(2*m-2*i)!)/ (i!*(m-i)!*(n-i)!*6^(n-i)), i=0..n); end;

%p [seq(f(3*n,2*n),n=0..10)];

%t Table[((6*n)!/(288)^n)*Hypergeometric1F1[-2*n,1/2-3*n,-3/2], {n,30}] (* _G. C. Greubel_, Oct 12 2023 *)

%o (Magma)

%o B:=Binomial;

%o A132202:= func< n | Factorial(6*n)/(288)^n*(&+[B(2*n,j)*B(3*n,j)*(-6)^j/(Factorial(j)*B(2*j,j)*B(6*n,2*j)): j in [0..2*n]]) >;

%o [A132202(n): n in [1..30]]; // _G. C. Greubel_, Oct 12 2023

%o (SageMath)

%o b=binomial

%o def A132202(n): return factorial(6*n)/(288)^n *simplify(hypergeometric([-2*n], [1/2-3*n], -3/2))

%o [A132202(n) for n in range(1,31)] # _G. C. Greubel_, Oct 12 2023

%Y Cf. A134648, A134772.

%K nonn,easy

%O 1,2

%A _Shanzhen Gao_, Nov 05 2007

%E Edited and extended with Maple code by _R. H. Hardin_ and _N. J. A. Sloane_, Oct 18 2009