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%I #23 Oct 22 2023 03:48:16
%S 1,518,15960915,4828311488700,6893870205562754400,
%T 32978529689054529966170400,428543560497255413435939747983950,
%U 13079873402738505705048288877402275168000,841990488872507644104617260743341546194585260000
%N 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}.
%H Vaclav Kotesovec, <a href="/A173791/b173791.txt">Table of n, a(n) for n = 1..88</a> (terms 1..50 from G. C. Greubel)
%F 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.
%F 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
%t 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]}]];
%t Table[a[n], {n, 12}] (* _G. C. Greubel_, Jul 13 2021 *)
%o (Sage)
%o B = binomial;
%o f = factorial;
%o @CachedFunction
%o 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] )
%o 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] )
%o [a(n) for n in (1..12)] # _G. C. Greubel_, Jul 13 2021
%Y Cf. A173789, A173790.
%K nonn
%O 1,2
%A _Shanzhen Gao_, Feb 24 2010
%E Edited by _G. C. Greubel_, Jul 13 2021
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