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A299039 Number of rooted trees with 2n nodes where each node has at most n children.

%I A299039 #22 Mar 17 2024 07:37:47
%S A299039 1,1,3,17,106,693,4690,32754,234746,1719325,12820920,97039824,
%T A299039 743680508,5759507657,45006692668,354425763797,2809931206626,
%U A299039 22409524536076,179655903886571,1447023307374888,11703779855021636,95020085240320710,774088021528328920
%N A299039 Number of rooted trees with 2n nodes where each node has at most n children.
%H A299039 Alois P. Heinz, <a href="/A299039/b299039.txt">Table of n, a(n) for n = 0..275</a>
%F A299039 a(n) = A299038(2n,n).
%F A299039 a(n) ~ c * d^n / n^(3/2), where d = A051491^2 = 8.736548423865419449938118272879... and c = A187770 / 2^(3/2) = 0.155536626247883986039760097126... - _Vaclav Kotesovec_, Feb 02 2018, updated Mar 17 2024
%e A299039 a(2) = 3:
%e A299039    o     o       o
%e A299039    |     |      / \
%e A299039    o     o     o   o
%e A299039    |    / \    |
%e A299039    o   o   o   o
%e A299039    |
%e A299039    o
%p A299039 b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
%p A299039       `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p A299039        b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
%p A299039     end:
%p A299039 a:= n-> `if`(n=0, 1, b(2*n-1$2, n$2)):
%p A299039 seq(a(n), n=0..25);
%t A299039 b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
%t A299039 a[n_] := If[n == 0, 1, b[2n - 1, 2n - 1, n, n]];
%t A299039 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 04 2018, from Maple *)
%Y A299039 Cf. A051491, A100034, A187770, A244407, A299038, A299098.
%K A299039 nonn
%O A299039 0,3
%A A299039 _Alois P. Heinz_, Feb 01 2018