A135589 - OEIS

%I #16 Feb 02 2024 09:06:47

%S 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,

%T 0,6,116,420,540,232,0,0,0,3,138,915,2160,2142,764,0,0,0,1,136,1605,

%U 6230,10766,8624,2620,0,0,0,0,116,2372,14436,39130,53312,35856,9496,0,0,0,0

%N 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.

%H Andrew Howroyd, <a href="/A135589/b135589.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%F 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

%e   1;

%e   0, 1;

%e   0, 0, 2;

%e   0, 0, 2,  4;

%e   0, 0, 1,  9,  10;

%e   0, 0, 0, 12,  36,  26;

%e   0, 0, 0, 10,  76, 140,  76;

%e   0, 0, 0,  6, 116, 420, 540, 232;

%e   ...

%o (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])))

%o { my(M=T(10)); for(i=1, #M, print(M[i,1..i])) } \\ _Andrew Howroyd_, Feb 01 2024

%Y Main diagonal gives A000085.

%Y Row sums give A135588.

%Y Column sums give A322661.

%K nonn,tabl

%O 0,6

%A _Vladeta Jovovic_, Feb 25 2008