A327827 - OEIS

%I #12 Dec 09 2020 15:06:11

%S 0,1,2,9,40,235,1476,11214,91848,859527,8710300,97675138,1179954612,

%T 15490520786,217602374458,3280028076615,52571985879600,

%U 895913825750191,16140560853800556,307048409240931810,6143666813617775100,129096480664676773542,2840750997343361802150

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

%H Alois P. Heinz, <a href="/A327827/b327827.txt">Table of n, a(n) for n = 0..450</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%F a(n) ~ c * n!, where c = A247551 = 2.5294774720791526481801161542539542411787... - _Vaclav Kotesovec_, Sep 28 2019

%p b:= proc(n, i, k) option remember; `if`(n=0, 1,

%p      `if`(i>n, 0, b(n, i+1, `if`(i=k, 0, k))+

%p      `if`(i=k, 0, b(n-i, i, k)*binomial(n, i))))

%p     end:

%p a:= n-> b(n, 1, 0)-b(n, 1$2):

%p seq(a(n), n=0..23);

%t 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 T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);

%t a[n_] := T[n, 1];

%t a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)

%Y Column k=1 of A327801.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 26 2019