A322115 Triangle read by rows where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.

1, 1, 1, 1, 2, 1, 1, 4, 4, 2, 1, 6, 11, 9, 3, 1, 9, 25, 34, 20, 6, 1, 12, 52, 104, 99, 49, 11, 1, 16, 94, 274, 387, 298, 118, 23, 1, 20, 162, 645, 1295, 1428, 881, 300, 47, 1, 25, 263, 1399, 3809, 5803, 5088, 2643, 765, 106, 1, 30, 407, 2823, 10187, 20645, 24606, 17872, 7878, 1998, 235

Offset: 0

Author

Gus Wiseman, Nov 26 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1
  1   1
  1   2   1
  1   4   4   2
  1   6  11   9   3
  1   9  25  34  20   6
  1  12  52 104  99  49  11
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274

Crossrefs

Row sums are A007719. Diagonal k = n-1 is A000055.

Cf. A000664, A007716, A007718, A191646, A191970, A275421, A317533, A322114.

Programs

PARI

EulerT(v)={my(p=exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1); Vec(p/x,-#v)}
InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v,x)={sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i],v[j])); g*x^(v[i]*v[j]/g))) + sum(i=1, #v, my(t=v[i]); ((t+1)\2)*x^t + if(t%2, 0, x^(t/2)))}
G(n,m)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(edges(p,x) + O(x*x^m), -m))); s/n!}
R(n)={Mat(apply(p->Col(p+O(y^n), -n), InvEulerMT(vector(n, k, 1 + y*Ser(G(k,n-1), y)))))}
{ my(T=R(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Nov 30 2018

Extensions

Terms a(28) and beyond from Andrew Howroyd, Nov 30 2018