A173791 a(n) is The number of (0,1)-matrices, A = (a_{ij}), of size (3n) X (2n) such that each row has exactly two 1's and each column has exactly three 1's and with the restriction that no 1 stands on the line from a_{11} to a_{22}.

1, 518, 15960915, 4828311488700, 6893870205562754400, 32978529689054529966170400, 428543560497255413435939747983950, 13079873402738505705048288877402275168000, 841990488872507644104617260743341546194585260000

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..88 (terms 1..50 from G. C. Greubel)

Formula

a(n) = Sum_{k=0..2n} ( Sum_{s=0..k} ( Sum_{j=0..2*n-k} ( Sum_{t=0..min(j, k-s)} ( (-1)^(k+s+j)*B(k, s)*B(2*n, k)*B(j, t)*B(2*n-k, j)*B(3*n-k, j)*j!*(6*n-k-2*j-s)!/(B(2*n-k, t)*2^(3*n-s-t)*6^(2*n-k-j+t)) )))), where B = binomial.

a(n) ~ sqrt(Pi) * 2^(n+1) * 3^(4*n + 1/2) * n^(6*n + 1/2) / exp(6*n+3). - Vaclav Kotesovec, Oct 21 2023

Mathematica

a[n_]:= a[n]= With[{B=Binomial}, Sum[(-1)^(k+s+j)*B[k, s]*B[2*n, k]*B[j, t]*B[2*n-k, j]*B[3*n-k, j]*j!*(6*n-k-2*j-s)!/(B[2*n-k, t]*2^(3*n-s-t)*6^(2*n-k-j+t)), {k, 0, 2*n}, {s, 0, k}, {j, 0, 2*n-k}, {t, 0, Min[j, k-s]}]];
Table[a[n], {n, 12}] (* G. C. Greubel, Jul 13 2021 *)

Prog

(Sage)
B = binomial;
f = factorial;
@CachedFunction
def c(n, k): return sum( sum( sum( (-1)^(s+j)*B(k, s)*B(j, t)*B(2*n-k, j)*B(3*n-k, j)*f(j)*f(6*n-k-2*j-s)*2^s*6^j/(B(2*n-k, t)*3^t) for t in [0..min(k-s, j)] ) for j in [0..2*n-k]) for s in [0..k]  )
def a(n): return sum( (-1)^k*B(2*n, k)*c(n, k)/(8^n*6^(2*n-k)) for k in [0..2*n] )
[a(n) for n in (1..12)] # G. C. Greubel, Jul 13 2021

Crossrefs

Cf. A173789, A173790.

Sequence in context: A236642 A253726 A293085 * A345587 A345845 A051983

Adjacent sequences: A173788 A173789 A173790 * A173792 A173793 A173794

Keyword

nonn

Author

Shanzhen Gao, Feb 24 2010

Extensions

Edited by G. C. Greubel, Jul 13 2021

Status

approved