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%I #28 Feb 12 2023 10:22:39
%S 1,1,1,2,4,9,20,48,115,286,719,1842,4765,12483,32964,87785,235305,
%T 634628,1720524,4686842,12820920,35206475,97010705,268154003,
%U 743351390,2066090876,5756490561,16074597300,44980514021,126109353817,354202275766,996517941454
%N Number of rooted unlabeled trees on n nodes where each node has at most 10 children.
%H Alois P. Heinz, <a href="/A292555/b292555.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>
%F Functional equation of G.f. is T(z) = z + z*Sum_{q=1..10} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is
%F T(z) = 1 + z*Z(S_10)(T(z)).
%F a(n) = Sum_{j=1..10} A244372(n,j) for n>0, a(0) = 1. - _Alois P. Heinz_, Sep 20 2017
%F a(n) / a(n+1) ~ 0.338329194566131211670667671160855741193081902868090986608524... - _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, 10$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, 10, 10]];
%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, A292553, A292554, A292556.
%Y Column k=10 of A299038.
%K nonn
%O 0,4
%A _Marko Riedel_, Sep 18 2017
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