A327827 Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating 1.

0, 1, 2, 9, 40, 235, 1476, 11214, 91848, 859527, 8710300, 97675138, 1179954612, 15490520786, 217602374458, 3280028076615, 52571985879600, 895913825750191, 16140560853800556, 307048409240931810, 6143666813617775100, 129096480664676773542, 2840750997343361802150

Offset: 0

Author

Alois P. Heinz, Sep 26 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• Wikipedia, Multinomial coefficients
• Wikipedia, Partition (number theory)

Crossrefs

Column k=1 of A327801.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1, `if`(i=k, 0, k))+
     `if`(i=k, 0, b(n-i, i, k)*binomial(n, i))))
    end:
a:= n-> b(n, 1, 0)-b(n, 1$2):
seq(a(n), n=0..23);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i], k]/i!]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
a[n_] := T[n, 1];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * n!, where c = A247551 = 2.5294774720791526481801161542539542411787... - Vaclav Kotesovec, Sep 28 2019