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

0,6

Links

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

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

Example

  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;
  ...

Prog

(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

Crossrefs

Main diagonal gives A000085.

Row sums give A135588.

Column sums give A322661.

Sequence in context: A343649 A195581 A020474 * A244312 A158122 A348165

Adjacent sequences: A135586 A135587 A135588 * A135590 A135591 A135592

Keyword

nonn,tabl

Author

Vladeta Jovovic, Feb 25 2008

Status

approved