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A035056 Number of asymmetric forests with n nodes.

%I A035056 #33 Dec 14 2024 05:27:26
%S A035056 1,1,0,0,0,0,0,1,2,4,9,21,44,96,206,450,981,2159,4757,10571,23563,
%T A035056 52835,118939,269047,610878,1392677,3186001,7313882,16842202,38900699,
%U A035056 90098260,209229601,487077685,1136549747,2657859059,6228447488,14624515804,34402798404
%N A035056 Number of asymmetric forests with n nodes.
%D A035056 Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 301 and 562.
%H A035056 Alois P. Heinz, <a href="/A035056/b035056.txt">Table of n, a(n) for n = 0..1000</a>
%H A035056 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A035056 <a href="/index/Fo#forests">Index entries for sequences related to forests</a>
%F A035056 Weigh transform of A000220.
%F A035056 a(n) ~ c * d^n / n^(5/2), where d = A246169 = 2.51754035263200389079535..., c = 0.421943694962576031011358... . - _Vaclav Kotesovec_, Aug 25 2014
%p A035056 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A035056       add(binomial(b((i-1)$2), j)*b(n-i*j, i-1), j=0..n/i)))
%p A035056     end:
%p A035056 g:= n-> b((n-1)$2):
%p A035056 h:= proc(n) option remember; g(n)-add(g(i)*g(n-i), i=0..n)/2
%p A035056        -`if`(irem(n, 2)=1, 0, g(n/2))/2
%p A035056     end:
%p A035056 f:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A035056       add(binomial(h(i), j)*f(n-i*j, i-1), j=0..n/i)))
%p A035056     end:
%p A035056 a:= n-> f(n, n):
%p A035056 seq(a(n), n=0..40);  # _Alois P. Heinz_, May 20 2013
%t A035056 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[b[i-1, i-1], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; g[n_] := b[n-1, n-1]; h[n_] := h[n] = g[n] - Sum[g[i]*g[n-i], {i, 0, n}]/2 - If[Mod[n, 2]==1, 0, g[n/2]]/2; f[n_, i_] := f[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[h[i], j]*f[n - i*j, i-1], {j, 0, n/i}]]]; a[n_] := f[n, n]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 21 2016, after _Alois P. Heinz_ *)
%Y A035056 Cf. A000220, A004111, A005195, A246169.
%K A035056 nonn
%O A035056 0,9
%A A035056 _Christian G. Bower_, Oct 15 1998