cp's OEIS Frontend

A266305 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.

%I A266305 #32 Jan 22 2024 08:45:45
%S A266305 1,1,7,74,1060,19013,408650,10219360,291158230,9302358947,
%T A266305 329192040880,12775809098058,539351216354728,24600280965461923,
%U A266305 1205263251360664310,63115789721408960624,3517483455875467926588,207834769804597591153769,12976002600530598793672490
%N A266305 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 A266305 Alois P. Heinz, <a href="/A266305/b266305.txt">Table of n, a(n) for n = 0..200</a>
%F A266305 a(n) = A138177(2n,n).
%e A266305 a(2) = 7:
%e A266305   [1 1]  [2 1]  [0 1]  [2 0]  [0 2]  [3 0]  [1 0]
%e A266305   [1 1]  [1 0]  [1 2]  [0 2]  [2 0]  [0 1]  [0 3].
%p A266305 gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
%p A266305 A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
%p A266305 a:= n-> add(A(2*n, n-j)*(-1)^j*binomial(n, j), j=0..n):
%p A266305 seq(a(n), n=0..20);
%t A266305 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 A266305 Cf. A138177, A268309.
%K A266305 nonn
%O A266305 0,3
%A A266305 _Alois P. Heinz_, Jan 31 2016