A343094 - OEIS

%I #12 Jan 02 2022 14:53:16

%S 1,1,4,24,201,2199,29879,486231,9228867,200272157,4892538679,

%T 132901744486,3974163191604,129735825879843,4591187990365503,

%U 175081375829138149,7157470516097359747,312267811360305253384,14481803617066951613463,711413177916751077562759

%N Number of connected loopless labeled multigraphs with n edges.

%H Andrew Howroyd, <a href="/A343094/b343094.txt">Table of n, a(n) for n = 0..200</a>

%t S[m_, n_] := Binomial[Binomial[m, 2] + n - 1, n];

%t R[nn_] := Module[{cc = Array[0&, {nn, nn}]}, cc[[1, 1]] = 1; For[m = 1, m <= nn, m++, For[n = 1, n <= nn - 1, n++, cc[[m, n + 1]] = S[m, n] - S[m - 1, n] - Sum[Sum[Binomial[m - 1, i - 1]*cc[[i, j + 1]]*S[m - i, n - j], {j, 1, n}], {i, 2, m - 1}]]]; cc // Transpose];

%t Total /@ R[20] (* _Jean-François Alcover_, Apr 15 2021, after _Andrew Howroyd_'s code for A290776 *)

%o (PARI)

%o Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}

%o seq(n)={Vec(vecsum(Connected(vector(2*n, j, 1/(1 - x + O(x*x^n))^binomial(j, 2)))))}

%Y Row sums of A290776.

%Y The unlabeled version is A050535.

%Y Cf. A322152.

%K nonn

%O 0,3

%A _Andrew Howroyd_, Apr 14 2021