%I #35 Feb 12 2023 10:22:23
%S 1,1,1,2,4,9,20,48,115,286,718,1839,4757,12460,32897,87592,234746,
%T 633013,1715851,4673320,12781759,35093010,96681705,267199518,
%U 740580555,2058042803,5733101603,16006590851,44782679547,125533577578,352525803976,991634575368
%N Number of rooted unlabeled trees on n nodes where each node has at most 8 children.
%H Alois P. Heinz, <a href="/A292553/b292553.txt">Table of n, a(n) for n = 0..1000</a>
%H Marko Riedel, <a href="https://math.stackexchange.com/questions/2434908/">Trees with bounded degree.</a>
%H Marko Riedel, <a href="/A292553/a292553.maple.txt">Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 1).</a>
%H Marko Riedel, <a href="/A292553/a292553_1.maple.txt">Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 2)</a>
%H Marko Riedel, <a href="/A292553/a292553_2.maple.txt">Maple code (FEQ 2) optimized for speed.</a>
%F Functional equation of G.f. is T(z) = z + z*Sum_{q=1..8} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is T(z) = 1 + z*Z(S_8)(T(z)).
%F a(n) = Sum_{j=1..8} A244372(n,j) for n>0, a(0) = 1. - _Alois P. Heinz_, Sep 20 2017
%F a(n) / a(n+1) ~ 0.338386042364849957035744926227166370702775721795018600630554... - _Robert A. Russell_, Feb 11 2023
%p b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
%p end:
%p a:= n-> `if`(n=0, 1, b(n-1$2, 8$2)):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Sep 20 2017
%t 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 a[n_] := If[n == 0, 1, b[n - 1, n - 1, 8, 8]];
%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292554, A292555, A292556.
%Y Column k=8 of A299038.
%K nonn
%O 0,4
%A _Marko Riedel_, Sep 18 2017