A266305 - OEIS

%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