A121227 - OEIS

%I #20 Jan 19 2025 11:29:28

%S 1,2,14,150,2210,41642,956878,25955630,811819826,28764498386,

%T 1138755852990,49817190098694,2386544217733906,124257113538066522,

%U 6986465328614267742,421887743219324342110,27231714819135144778722,1871047822756547798671074

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

%H Seiichi Manyama, <a href="/A121227/b121227.txt">Table of n, a(n) for n = 0..364</a> (terms 0..75 from Nathaniel Johnston)

%F 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

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

%o (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

%Y Row n=2 of A331315.

%Y Unlabeled analog is A007717.

%K easy,nonn

%O 0,2

%A _Vladeta Jovovic_, Sep 06 2006