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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. 15
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1325 EXAMPLE 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}} PROG (PARI) InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i) )} 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 CROSSREFS Row sums are A191970. Last column is A000055. Cf. A000664, A007716, A007718, A007719, A054923, A191646, A275421, A317533, A321254. Sequence in context: A055654 A170849 A292260 * A062787 A131370 A261180 Adjacent sequences:  A322111 A322112 A322113 * A322115 A322116 A322117 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Nov 26 2018 EXTENSIONS Terms a(28) and beyond from Andrew Howroyd, Oct 22 2019 STATUS approved

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Last modified January 23 19:36 EST 2020. Contains 331175 sequences. (Running on oeis4.)