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.

1, 1, 7, 74, 1060, 19013, 408650, 10219360, 291158230, 9302358947, 329192040880, 12775809098058, 539351216354728, 24600280965461923, 1205263251360664310, 63115789721408960624, 3517483455875467926588, 207834769804597591153769, 12976002600530598793672490

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) = A138177(2n,n).

Example

a(2) = 7:
  [1 1]  [2 1]  [0 1]  [2 0]  [0 2]  [3 0]  [1 0]
  [1 1]  [1 0]  [1 2]  [0 2]  [2 0]  [0 1]  [0 3].

Maple

gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
a:= n-> add(A(2*n, n-j)*(-1)^j*binomial(n, j), j=0..n):
seq(a(n), n=0..20);

Mathematica

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 *)

Crossrefs

Cf. A138177, A268309.

Sequence in context: A295245 A365844 A341330 * A098118 A097821 A337387

Adjacent sequences: A266302 A266303 A266304 * A266306 A266307 A266308

Keyword

nonn

Author

Alois P. Heinz, Jan 31 2016

Status

approved