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