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.

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, 10, 61, 112, 81, 23, 0, 0, 0, 5, 75, 262, 347, 204, 47, 0, 0, 0, 2, 68, 463, 1059, 1085, 526, 106, 0, 0, 0, 1, 49, 625, 2458, 4091, 3348, 1376, 235

Offset: 0

Author

Gus Wiseman, Nov 26 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   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
Non-isomorphic representatives of the graphs counted in row 4:
  {{2}{3}{12}{13}}   {{4}{12}{23}{34}}   {{13}{24}{35}{45}}
  {{2}{3}{13}{23}}   {{4}{13}{23}{34}}   {{14}{25}{35}{45}}
  {{3}{12}{13}{23}}  {{4}{13}{24}{34}}   {{15}{25}{35}{45}}
                     {{4}{14}{24}{34}}
                     {{12}{13}{24}{34}}
                     {{14}{23}{24}{34}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325

Crossrefs

Row sums are A191970. Last column is A000055.

Cf. A000664, A007716, A007718, A007719, A054923, A191646, A275421, A317533, A321254.

Programs

PARI

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,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)))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p,i->1+x^i)); s/n!}
T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
{my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ Andrew Howroyd, Oct 22 2019

Extensions

Terms a(28) and beyond from Andrew Howroyd, Oct 22 2019