A121227 Number of labeled multigraphs with loops and with n edges and no vertex of degree 0.

1, 2, 14, 150, 2210, 41642, 956878, 25955630, 811819826, 28764498386, 1138755852990, 49817190098694, 2386544217733906, 124257113538066522, 6986465328614267742, 421887743219324342110, 27231714819135144778722, 1871047822756547798671074

Offset: 0

Author

Vladeta Jovovic, Sep 06 2006

Keywords

• easy
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..364 (terms 0..75 from Nathaniel Johnston)

Crossrefs

Row n=2 of A331315.

Unlabeled analog is A007717.

Programs

Maple

seq(sum(binomial(m*(m+1)/2+n-1,n)/2^(m+1),m=0..infinity),n=0..10);

PARI

a(n)={sum(k=0, 2*n, binomial(k*(k+1)/2+n-1, n)*sum(r=k, 2*n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 15 2018

Formula

a(n) = Sum_{k=0..2*n} binomial(k*(k+1)/2+n-1, n)*(Sum_{r=k..2*n} binomial(r, k)*(-1)^(r-k)). - Andrew Howroyd, Sep 15 2018