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A055192 Number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block, up to isomorphism.

%I A055192 #19 Sep 05 2019 08:35:07
%S A055192 1,2,5,12,35,108,393,1666,8543,54190,436740,4565450,62930604,
%T A055192 1156277748,28509174012,946786816168,42448800498744,2573207315483554,
%U A055192 211180300735118954,23490473719472829824,3545759835559406756008,727077827560669587718290
%N A055192 Number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block, up to isomorphism.
%C A055192 Also the number of connected split graphs on n vertices (cf. A048194). - _Falk Hüffner_, Dec 01 2015
%C A055192 Inverse Euler transform is A007776. - _Andrew Howroyd_, Oct 03 2018
%H A055192 Alois P. Heinz, <a href="/A055192/b055192.txt">Table of n, a(n) for n = 2..40</a>
%t A055192 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 A055192 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 A055192 A[n_, k_] := g[Min[n, k], Abs[n - k]];
%t A055192 A049312[d_] := Sum[A[n, d - n], {n, 0, d}];
%t A055192 Differences[Table[A049312[n], {n, 0, 23}], 2] (* _Jean-François Alcover_, Sep 05 2019, after _Alois P. Heinz_ in A049312 *)
%Y A055192 Equals second differences of A049312.
%Y A055192 Cf. A007776, A024206, A055609, A055082, A055083, A055084.
%Y A055192 Row sums of A056152 and also of A122083.
%K A055192 nonn
%O A055192 2,2
%A A055192 _Vladeta Jovovic_, Jun 18 2000