

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}.


2



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

1,2


LINKS



FORMULA

a(n) = Sum_{k=0..2n} ( Sum_{s=0..k} ( Sum_{j=0..2*nk} ( Sum_{t=0..min(j, ks)} ( (1)^(k+s+j)*B(k, s)*B(2*n, k)*B(j, t)*B(2*nk, j)*B(3*nk, j)*j!*(6*nk2*js)!/(B(2*nk, t)*2^(3*nst)*6^(2*nkj+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*nk, j]*B[3*nk, j]*j!*(6*nk2*js)!/(B[2*nk, t]*2^(3*nst)*6^(2*nkj+t)), {k, 0, 2*n}, {s, 0, k}, {j, 0, 2*nk}, {t, 0, Min[j, ks]}]];


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*nk, j)*B(3*nk, j)*f(j)*f(6*nk2*js)*2^s*6^j/(B(2*nk, t)*3^t) for t in [0..min(ks, j)] ) for j in [0..2*nk]) for s in [0..k] )
def a(n): return sum( (1)^k*B(2*n, k)*c(n, k)/(8^n*6^(2*nk)) for k in [0..2*n] )


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



