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A036721 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(5), A(x)).

%I A036721 #38 Jul 08 2025 21:54:52
%S A036721 1,1,1,2,4,9,20,47,112,277,693,1766,4547,11852,31146,82534,220149,
%T A036721 590834,1593951,4320723,11761394,32138301,88121176,242383729,
%U A036721 668607115,1849194691,5126800907,14245679652,39666239726,110661514973,309280533011,865839831118
%N A036721 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(5), A(x)).
%C A036721 Also the number of rooted trees where each node has at most 5 children. [_Patrick Devlin_, Apr 30 2012]
%H A036721 Alois P. Heinz, <a href="/A036721/b036721.txt">Table of n, a(n) for n = 0..1000</a>
%H A036721 Joerg Arndt, <a href="/A036721/a036721_1.txt">The a(6) = 20 rooted trees with 5 nodes and out-degrees <= 5</a>
%F A036721 a(n) = Sum_{j=1..5} A244372(n,j) for n>0, a(0) = 1. - _Alois P. Heinz_, Sep 19 2017
%F A036721 a(n) / a(n+1) ~ 0.340017469151060086823930137816585262710976835711484267209811... - _Robert A. Russell_, Feb 11 2023
%p A036721 b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
%p A036721       `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p A036721        b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
%p A036721     end:
%p A036721 a:= n-> `if`(n=0, 1, b(n-1$2, 5$2)):
%p A036721 seq(a(n), n=0..35);  # _Alois P. Heinz_, Sep 20 2017
%t A036721 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 A036721 a[n_] := If[n == 0, 1, b[n - 1, n - 1, 5, 5]];
%t A036721 Table[a[n], {n, 0, 35}] // Flatten (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A036721 Cf. A000081, A036717, A036718, A036719, A036720, A036722, A182378, A244372, A292553, A292554, A292555, A292556.
%Y A036721 Column k=5 of A299038.
%K A036721 nonn
%O A036721 0,4
%A A036721 _N. J. A. Sloane_