A135589 Triangle T(n,k) read by rows: number of k X k symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.

1, 0, 1, 0, 0, 2, 0, 0, 2, 4, 0, 0, 1, 9, 10, 0, 0, 0, 12, 36, 26, 0, 0, 0, 10, 76, 140, 76, 0, 0, 0, 6, 116, 420, 540, 232, 0, 0, 0, 3, 138, 915, 2160, 2142, 764, 0, 0, 0, 1, 136, 1605, 6230, 10766, 8624, 2620, 0, 0, 0, 0, 116, 2372, 14436, 39130, 53312, 35856, 9496, 0, 0, 0, 0

Offset: 0

Author

Vladeta Jovovic, Feb 25 2008

Keywords

• nonn
• tabl

Examples

			  1;
  0, 1;
  0, 0, 2;
  0, 0, 2,  4;
  0, 0, 1,  9,  10;
  0, 0, 0, 12,  36,  26;
  0, 0, 0, 10,  76, 140,  76;
  0, 0, 0,  6, 116, 420, 540, 232;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Main diagonal gives A000085.

Row sums give A135588.

Column sums give A322661.

Programs

PARI

T(n)=my(A=O(x*x^n), v=vector(n+1, k, k--;Col(A+(1+x+A)^k*(1+x^2+A)^binomial(k,2)))); Mat(vector(n+1, k, k--; sum(j=0, k, (-1)^(k-j)*binomial(k,j)*v[1+j])))
{ my(M=T(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Feb 01 2024

Formula

G.f. of column k: Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (1 + x)^j * (1 + x^2)^binomial(j,2). - Andrew Howroyd, Feb 01 2024