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%I #33 Feb 12 2023 10:22:16
%S 1,1,1,2,4,9,20,48,115,285,716,1833,4740,12410,32754,87176,233547,
%T 629540,1705809,4644231,12697500,34848694,95973026,265142431,
%U 734606478,2040683413,5682634446,15859800889,44355531103,124290064228,348904212741,981082979409
%N G.f. satisfies A(x) = 1 + x*cycle_index(Sym(7), A(x)).
%C Number of rooted trees where each node has at most 7 children.
%H Alois P. Heinz, <a href="/A182378/b182378.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{j=1..7} A244372(n,j) for n>0, a(0) = 1. - _Alois P. Heinz_, Sep 19 2017
%F a(n) / a(n+1) ~ 0.338512011286603947719604869750539045616436718225097926729820... - _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, 7$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, 7, 7]];
%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jan 15 2018, after _Alois P. Heinz_ *)
%Y Cf. A000081, A036717, A036718, A036719, A036720, A036721, A036722, A244372, A292553, A292554, A292555, A292556.
%Y Column k=7 of A299038.
%K nonn
%O 0,4
%A _Michael Burkhart_, Apr 26 2012
%E More terms from _Patrick Devlin_, Apr 29 2012
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