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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.

%I A322115 #16 May 18 2023 01:55:39
%S A322115 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,
%T A322115 16,94,274,387,298,118,23,1,20,162,645,1295,1428,881,300,47,1,25,263,
%U A322115 1399,3809,5803,5088,2643,765,106,1,30,407,2823,10187,20645,24606,17872,7878,1998,235
%N A322115 Triangle read by rows where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.
%H A322115 Andrew Howroyd, <a href="/A322115/b322115.txt">Table of n, a(n) for n = 0..1274</a>
%e A322115 Triangle begins:
%e A322115   1
%e A322115   1   1
%e A322115   1   2   1
%e A322115   1   4   4   2
%e A322115   1   6  11   9   3
%e A322115   1   9  25  34  20   6
%e A322115   1  12  52 104  99  49  11
%o A322115 (PARI)
%o A322115 EulerT(v)={my(p=exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1); Vec(p/x,-#v)}
%o A322115 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))}
%o A322115 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}
%o A322115 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)))}
%o A322115 G(n,m)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(edges(p,x) + O(x*x^m), -m))); s/n!}
%o A322115 R(n)={Mat(apply(p->Col(p+O(y^n), -n), InvEulerMT(vector(n, k, 1 + y*Ser(G(k,n-1), y)))))}
%o A322115 { my(T=R(10)); for(n=1, #T, print(T[n, 1..n])) } \\ _Andrew Howroyd_, Nov 30 2018
%Y A322115 Row sums are A007719. Diagonal k = n-1 is A000055.
%Y A322115 Cf. A000664, A007716, A007718, A191646, A191970, A275421, A317533, A322114.
%K A322115 nonn,tabl
%O A322115 0,5
%A A322115 _Gus Wiseman_, Nov 26 2018
%E A322115 Terms a(28) and beyond from _Andrew Howroyd_, Nov 30 2018