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A292554 Number of rooted unlabeled trees on n nodes where each node has at most 9 children.

%I A292554 #27 Feb 12 2023 10:22:31
%S A292554 1,1,1,2,4,9,20,48,115,286,719,1841,4763,12477,32947,87735,235162,
%T A292554 634212,1719325,4683368,12810871,35177357,96926335,267909285,
%U A292554 742641309,2064029034,5750500663,16057186086,44929879114,125962026154,353773417487,995269027339
%N A292554 Number of rooted unlabeled trees on n nodes where each node has at most 9 children.
%H A292554 Alois P. Heinz, <a href="/A292554/b292554.txt">Table of n, a(n) for n = 0..1000</a>
%H A292554 Marko Riedel, <a href="https://math.stackexchange.com/questions/2434908/">Trees with bounded degree.</a>
%F A292554 Functional equation of G.f. is T(z) = z + z*Sum_{q=1..9} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is
%F A292554 T(z) = 1 + z*Z(S_9)(T(z)).
%F A292554 a(n) = Sum_{j=1..9} A244372(n,j) for n>0, a(0) = 1. - _Alois P. Heinz_, Sep 20 2017
%F A292554 a(n) / a(n+1) ~ 0.338343552789108712866488147828528012266693326385052387884853... - _Robert A. Russell_, Feb 11 2023
%p A292554 b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
%p A292554       `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p A292554        b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
%p A292554     end:
%p A292554 a:= n-> `if`(n=0, 1, b(n-1$2, 9$2)):
%p A292554 seq(a(n), n=0..35);  # _Alois P. Heinz_, Sep 20 2017
%t A292554 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 A292554 a[n_] := If[n == 0, 1, b[n - 1, n - 1, 9, 9]];
%t A292554 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A292554 Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292553, A292555, A292556.
%Y A292554 Column k=9 of A299038.
%K A292554 nonn
%O A292554 0,4
%A A292554 _Marko Riedel_, Sep 18 2017