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A268309 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 n^2.

%I A268309 #13 Jan 22 2024 08:46:08
%S A268309 1,1,7,347,83785,85813461,362302219609,6227015262941276,
%T A268309 433865390872310453097,122285854086662347886884837,
%U A268309 139236232279790897112737794283927,639720298831885406784643598607618757713,11848024220605180271987429760766015754937928643
%N A268309 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 n^2.
%H A268309 Alois P. Heinz, <a href="/A268309/b268309.txt">Table of n, a(n) for n = 0..45</a>
%F A268309 a(n) = A138177(n^2,n).
%e A268309 a(2) = 7:
%e A268309   [1 1]  [2 1]  [0 1]  [2 0]  [0 2]  [3 0]  [1 0]
%e A268309   [1 1]  [1 0]  [1 2]  [0 2]  [2 0]  [0 1]  [0 3].
%p A268309 gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
%p A268309 A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
%p A268309 a:= n-> add(A(n^2, n-j)*(-1)^j*binomial(n, j), j=0..n):
%p A268309 seq(a(n), n=0..15);
%t A268309 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[n^2, n-j]*(-1)^j*Binomial[n, j], {j, 0, n}]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Feb 25 2017, translated from Maple *)
%Y A268309 Cf. A138177, A266305.
%K A268309 nonn
%O A268309 0,3
%A A268309 _Alois P. Heinz_, Jan 31 2016