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A215930 Number of forests on unlabeled nodes with n edges and no single node trees.

%I A215930 #37 May 11 2017 16:51:15
%S A215930 1,1,2,4,8,16,34,71,154,341,768,1765,4134,9838,23766,58226,144353,
%T A215930 361899,916152,2339912,6023447,15617254,40752401,106967331,282267774,
%U A215930 748500921,1993727506,5332497586,14316894271,38574473086,104273776038,282733466684,768809041078
%N A215930 Number of forests on unlabeled nodes with n edges and no single node trees.
%C A215930 Each forest counted by a(n) with n>0 has number of nodes from the interval [n+1,2*n] and number of trees in [1,n].
%C A215930 Also limiting sequence of reversed rows of A095133.
%C A215930 Differs from A011782 first at n=6 (32) and from A088325 at n=8 (153).
%H A215930 Alois P. Heinz, <a href="/A215930/b215930.txt">Table of n, a(n) for n = 0..650</a>
%F A215930 a(n) = A095133(2*n,n).
%F A215930 a(n) = A105821(2*n+1,n+1). - _Alois P. Heinz_, Jul 10 2013
%F A215930 a(n) = A136605(2*n+1,n). - _Alois P. Heinz_, Apr 11 2014
%F A215930 a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.955765285..., c = 3.36695186... . - _Vaclav Kotesovec_, Sep 10 2014
%e A215930 a(0) = 1: (  ), the empty forest with 0 trees and 0 edges.
%e A215930 a(1) = 1: ( o-o ), 1 tree and 1 edge.                      o
%e A215930 a(2) = 2: ( o-o-o ), ( o-o o-o ).                          |
%e A215930 a(3) = 4: ( o-o-o-o ), ( o-o-o o-o ), ( o-o o-o o-o ), ( o-o-o ).
%p A215930 with(numtheory):
%p A215930 b:= proc(n) option remember; local d, j; `if`(n<=1, n,
%p A215930       (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
%p A215930     end:
%p A215930 t:= proc(n) option remember; local k; `if` (n=0, 1, b(n)-
%p A215930       (add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
%p A215930     end:
%p A215930 g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
%p A215930       `if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j)*
%p A215930        binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
%p A215930     end:
%p A215930 a:= n-> g(2*n, 2*n, n):
%p A215930 seq(a(n), n=0..40);
%t A215930 nn = 30; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
%t A215930 a[1] = 1; sol =
%t A215930 SolveAlways[
%t A215930   0 == Series[
%t A215930     t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];
%t A215930 b[x_] := Sum[a[n] x^n /. sol, {n, 0, nn}]; ft =
%t A215930 Drop[Flatten[
%t A215930    CoefficientList[Series[b[x] - (b[x]^2 - b[x^2])/2, {x, 0, nn}],
%t A215930     x]], 1]; Drop[
%t A215930 CoefficientList[
%t A215930   Series[Product[1/(1 - y ^(i - 1))^ft[[i]], {i, 2, nn}], {y, 0, nn}],
%t A215930 y], -1] (* _Geoffrey Critzer_, Nov 10 2014 *)
%Y A215930 Cf. A000055, A005195, A011782, A051491, A088325, A095133, A105821, A136605.
%K A215930 nonn
%O A215930 0,3
%A A215930 _Alois P. Heinz_, Aug 27 2012