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).

1, 1, 2, 6, 18, 90, 414, 2892, 18342, 155124, 1265130, 13413240, 129656286, 1564538796, 18285385518, 255345207156, 3378398348214, 52931303772912, 797460543143154, 13926097774972152, 234050020177159926, 4466082284967035124, 83159771376289666806

Offset: 1

Author

Alois P. Heinz, Sep 23 2019

Keywords

• nonn

Comments

The formula is a generalization of the formula given in A327643.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..460
• Wikipedia, Multinomial coefficients
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A327643, A327711.

Programs

Maple

with(combinat):
a:= proc(n) option remember; `if`(n<2, 1, add(mul(a(i), i=p)
      *multinomial(n-nops(p), map(x-> x-1, p)[]),
       p=select(x-> nops(x)>1, partition(n))))
    end:
seq(a(n), n=1..24);
# second Maple program:
b:= proc(n, p, i) option remember; `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)!))
    end:
a:= n-> `if`(n<2, 1, b(n$2, n-1)):
seq(a(n), n=1..24);

Mathematica

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)!]];
a[n_] := If[n < 2, 1, b[n, n, n - 1]];
Array[a, 24] (* Jean-François Alcover, May 03 2020, after 2nd Maple program *)