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A322115
Triangle read by rows where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.
12
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
0,5
LINKS
EXAMPLE
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
PROG
(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
CROSSREFS
Row sums are A007719. Diagonal k = n-1 is A000055.
Sequence in context: A350012 A322038 A123521 * A294217 A123246 A122518
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 26 2018
EXTENSIONS
Terms a(28) and beyond from Andrew Howroyd, Nov 30 2018
STATUS
approved