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A215979 Number of simple unlabeled graphs on 2*n nodes with exactly n connected components that are trees or cycles.

%I A215979 #17 Feb 26 2017 05:43:27
%S A215979 1,1,3,6,13,26,56,115,247,533,1175,2636,6040,14078,33401,80524,196897,
%T A215979 487781,1222279,3094507,7905992,20364597,52838720,138001953,362565398,
%U A215979 957687474,2542056376,6777855755,18146153182,48766704695,131517773945,355842838357
%N A215979 Number of simple unlabeled graphs on 2*n nodes with exactly n connected components that are trees or cycles.
%C A215979 Limiting sequence of reversed rows of A215977.  Also central elements of rows of A215977.
%H A215979 Alois P. Heinz, <a href="/A215979/b215979.txt">Table of n, a(n) for n = 0..200</a>
%F A215979 a(n) = A215977(2*n,n).
%F A215979 a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856..., c = 4.034813602... . - _Vaclav Kotesovec_, Aug 31 2014
%e A215979 a(3) = 6: .o-o o.  .o-o o.  .o-o o.  .o-o o.  .o-o o.  .o o o.
%e A215979           .| |  .  .|    .  .|\   .  .|/   .  .|    .  .| | |.
%e A215979           .o-o o.  .o-o o.  .o o o.  .o o-o.  .o o-o.  .o o o.
%p A215979 with(numtheory):
%p A215979 b:= proc(n) option remember; local d, j; `if`(n<=1, n,
%p A215979       (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
%p A215979     end:
%p A215979 g:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
%p A215979       (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
%p A215979     end:
%p A215979 p:= proc(n, i, t) option remember; `if`(n<t, 0, `if`(n=t, 1,
%p A215979       `if`(min(i, t)<1, 0, add(binomial(g(i)+j-1, j)*
%p A215979        p(n-i*j, i-1, t-j), j=0..min(n/i,t)))))
%p A215979     end:
%p A215979 a:= n-> p(2*n, 2*n, n):
%p A215979 seq(a(n), n=0..35);
%t A215979 b[n_] := b[n] = If[n <= 1, n, (Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}])/(n-1)];
%t A215979 g[n_] := g[n] = If[n > 2, 1, 0] + b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[EvenQ[n], b[n/2], 0])/2;
%t A215979 p[n_, i_, t_] := p[n, i, t] = If[n < t, 0, If[n == t, 1, If[Min[i, t]<1, 0, Sum[Binomial[g[i]+j-1, j]*p[n-i*j, i-1, t-j], {j, 0, Min[n/i, t]}]]]];
%t A215979 a[n_] := p[2*n, 2*n, n];
%t A215979 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Feb 26 2017, after _Alois P. Heinz_ *)
%Y A215979 Cf. A215977, A051491.
%K A215979 nonn
%O A215979 0,3
%A A215979 _Alois P. Heinz_, Aug 29 2012