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A318870 Number of connected bipartite graphs on n unlabeled nodes with a distinguished bipartite block.

%I A318870 #15 Oct 03 2018 19:14:04
%S A318870 1,2,1,2,4,10,27,88,328,1460,7799,51196,422521,4483460,62330116,
%T A318870 1150504224,28434624153,945480850638,42417674401330,2572198227615998,
%U A318870 211135833162079184,23487811567341121158,3545543330739039981738,727053904070651775719646
%N A318870 Number of connected bipartite graphs on n unlabeled nodes with a distinguished bipartite block.
%C A318870 Essentially the same as A007776. - _Georg Fischer_, Oct 02 2018
%H A318870 Andrew Howroyd, <a href="/A318870/b318870.txt">Table of n, a(n) for n = 0..50</a>
%F A318870 Inverse Euler transform of A049312.
%e A318870 a(1) = 2 because the single node can either be in the distinguished bipartite block or not.
%e A318870 a(2) = 1 because the only connected bipartite graph on two nodes is the complete graph on two nodes.
%e A318870 a(3) = 2 because the only connected bipartite graph on three nodes is the path graph on three nodes and there is a choice about which nodes are in the distinguished block.
%t A318870 mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t A318870 EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
%t A318870 b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A318870 g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
%t A318870 A[n_, k_] := g[Min[n, k], Abs[n - k]];
%t A318870 b[d_] := Sum[A[n, d - n], {n, 0, d}];
%t A318870 Join[{1}, EULERi[Array[b, 23]]] (* _Jean-François Alcover_, Sep 13 2018, after _Alois P. Heinz_ in A049312 *)
%Y A318870 Cf. A005142, A049312, A123549, A318869.
%K A318870 nonn
%O A318870 0,2
%A A318870 _Andrew Howroyd_, Sep 04 2018