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A327729 a(n) = Sum_{p} M(n-k; p_1-1, ..., p_k-1) * Product_{j=1..k} a(p_j), where p = (p_1, ..., p_k) ranges over all partitions of n into smaller parts (k is a partition length and M is a multinomial).

%I A327729 #16 May 03 2020 13:42:18
%S A327729 1,1,2,6,18,90,414,2892,18342,155124,1265130,13413240,129656286,
%T A327729 1564538796,18285385518,255345207156,3378398348214,52931303772912,
%U A327729 797460543143154,13926097774972152,234050020177159926,4466082284967035124,83159771376289666806
%N A327729 a(n) = Sum_{p} M(n-k; p_1-1, ..., p_k-1) * Product_{j=1..k} a(p_j), where p = (p_1, ..., p_k) ranges over all partitions of n into smaller parts (k is a partition length and M is a multinomial).
%C A327729 The formula is a generalization of the formula given in A327643.
%H A327729 Alois P. Heinz, <a href="/A327729/b327729.txt">Table of n, a(n) for n = 1..460</a>
%H A327729 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A327729 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%p A327729 with(combinat):
%p A327729 a:= proc(n) option remember; `if`(n<2, 1, add(mul(a(i), i=p)
%p A327729       *multinomial(n-nops(p), map(x-> x-1, p)[]),
%p A327729        p=select(x-> nops(x)>1, partition(n))))
%p A327729     end:
%p A327729 seq(a(n), n=1..24);
%p A327729 # second Maple program:
%p A327729 b:= proc(n, p, i) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A327729       b(n, p, i-1) +a(i)*b(n-i, p-1, min(n-i, i))/(i-1)!))
%p A327729     end:
%p A327729 a:= n-> `if`(n<2, 1, b(n$2, n-1)):
%p A327729 seq(a(n), n=1..24);
%t A327729 b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i < 1, 0, b[n, p, i - 1] + a[i] b[n - i, p - 1, Min[n - i, i]]/(i - 1)!]];
%t A327729 a[n_] := If[n < 2, 1, b[n, n, n - 1]];
%t A327729 Array[a, 24] (* _Jean-François Alcover_, May 03 2020, after 2nd Maple program *)
%Y A327729 Cf. A327643, A327711.
%K A327729 nonn
%O A327729 1,3
%A A327729 _Alois P. Heinz_, Sep 23 2019