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%I #32 Jan 22 2024 08:45:45
%S 1,1,7,74,1060,19013,408650,10219360,291158230,9302358947,
%T 329192040880,12775809098058,539351216354728,24600280965461923,
%U 1205263251360664310,63115789721408960624,3517483455875467926588,207834769804597591153769,12976002600530598793672490
%N Number of n X n symmetric matrices with nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to 2n.
%H Alois P. Heinz, <a href="/A266305/b266305.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = A138177(2n,n).
%e a(2) = 7:
%e [1 1] [2 1] [0 1] [2 0] [0 2] [3 0] [1 0]
%e [1 1] [1 0] [1 2] [0 2] [2 0] [0 1] [0 3].
%p gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
%p A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
%p a:= n-> add(A(2*n, n-j)*(-1)^j*binomial(n, j), j=0..n):
%p seq(a(n), n=0..20);
%t gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); A[n_, k_] := SeriesCoefficient[ gf[k], {x, 0, n}]; a[n_] := Sum[A[2*n, n-j]*(-1)^j*Binomial[n, j], {j, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 25 2017, translated from Maple *)
%Y Cf. A138177, A268309.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jan 31 2016
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