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A036722 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(6), A(x)).

%I A036722 #31 Jul 08 2025 21:54:59
%S A036722 1,1,1,2,4,9,20,48,114,283,710,1816,4690,12267,32338,85978,230080,
%T A036722 619521,1676808,4560286,12454272,34143682,93928091,259208006,
%U A036722 717375068,1990625390,5537142610,15436744525,43124847431,120708508008,338477040445,950714584576
%N A036722 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(6), A(x)).
%C A036722 a(n) is also the number of rooted trees where each node has at most 6 children. [_Patrick Devlin_, Apr 29 2012]
%H A036722 Alois P. Heinz, <a href="/A036722/b036722.txt">Table of n, a(n) for n = 0..1000</a>
%F A036722 a(n) = Sum_{j=1..6} A244372(n,j) for n>0, a(0) = 1. - _Alois P. Heinz_, Sep 19 2017
%F A036722 a(n) / a(n+1) ~ 0.338887196052856714304749078960983936661485522864792573284374... - _Robert A. Russell_, Feb 11 2023
%p A036722 b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
%p A036722       `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p A036722        b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
%p A036722     end:
%p A036722 a:= n-> `if`(n=0, 1, b(n-1$2, 6$2)):
%p A036722 seq(a(n), n=0..35);  # _Alois P. Heinz_, Sep 20 2017
%t A036722 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 A036722 a[n_] := If[n == 0, 1, b[n - 1, n - 1, 6, 6]];
%t A036722 Table[a[n] , {n, 0, 35}] // Flatten (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A036722 Cf. A000081, A036717, A036718, A036719, A036720, A036721, A182378, A244372, A292553, A292554, A292555, A292556.
%Y A036722 Column k=6 of A299038.
%K A036722 nonn
%O A036722 0,4
%A A036722 _N. J. A. Sloane_