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A322114 Regular triangle read by rows where T(n,k) is the number of unlabeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

%I A322114 #10 Mar 23 2023 23:08:44
%S A322114 1,1,1,0,1,1,0,1,3,2,0,0,3,6,3,0,0,2,11,14,6,0,0,1,13,35,33,11,0,0,0,
%T A322114 10,61,112,81,23,0,0,0,5,75,262,347,204,47,0,0,0,2,68,463,1059,1085,
%U A322114 526,106,0,0,0,1,49,625,2458,4091,3348,1376,235
%N A322114 Regular triangle read by rows where T(n,k) is the number of unlabeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.
%H A322114 Andrew Howroyd, <a href="/A322114/b322114.txt">Table of n, a(n) for n = 0..1325</a>
%e A322114 Triangle begins:
%e A322114    1
%e A322114    1   1
%e A322114    0   1   1
%e A322114    0   1   3   2
%e A322114    0   0   3   6   3
%e A322114    0   0   2  11  14   6
%e A322114    0   0   1  13  35  33  11
%e A322114 Non-isomorphic representatives of the graphs counted in row 4:
%e A322114   {{2}{3}{12}{13}}   {{4}{12}{23}{34}}   {{13}{24}{35}{45}}
%e A322114   {{2}{3}{13}{23}}   {{4}{13}{23}{34}}   {{14}{25}{35}{45}}
%e A322114   {{3}{12}{13}{23}}  {{4}{13}{24}{34}}   {{15}{25}{35}{45}}
%e A322114                      {{4}{14}{24}{34}}
%e A322114                      {{12}{13}{24}{34}}
%e A322114                      {{14}{23}{24}{34}}
%o A322114 (PARI)
%o A322114 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 A322114 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 A322114 edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
%o A322114 G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p,i->1+x^i)); s/n!}
%o A322114 T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
%o A322114 {my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ _Andrew Howroyd_, Oct 22 2019
%Y A322114 Row sums are A191970. Last column is A000055.
%Y A322114 Cf. A000664, A007716, A007718, A007719, A054923, A191646, A275421, A317533, A321254.
%K A322114 nonn,tabl
%O A322114 0,9
%A A322114 _Gus Wiseman_, Nov 26 2018
%E A322114 Terms a(28) and beyond from _Andrew Howroyd_, Oct 22 2019