%I #15 Mar 18 2024 06:01:40

%S 0,0,0,3,16,125,711,5915,46264,438681,4371085,49321745,588219523,

%T 7751724513,108240044745,1633289839823,26102966544024,445098171557393,

%U 8006283582196761,152353662601600853,3046062181913575921,64015245150903376151,1408108698825029286195

%N Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size two.

%H Alois P. Heinz, <a href="/A327826/b327826.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 = Sum_{k>=2} 1/(k! - 1) = A331373 = 1.253498755699953471643360937905798940369232208332... - _Vaclav Kotesovec_, Sep 28 2019, updated Jul 19 2021

%p with(combinat):

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

%p `if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*

%p multinomial(n, n-i*j, i$j), j=0..n/i))), x, 3)

%p end:

%p a:= n-> coeff(b(n$2), x, 2):

%p seq(a(n), n=0..25);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, Sum[x^Sign[j] b[n - i*j, i - 1] multinomial[n, Join[{n - i*j}, Table[i, {j}]]], {j, 0, n/i}]]], {x, 0, 3}];

%t a[n_] := SeriesCoefficient[b[n, n], {x, 0, 2}];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)

%Y Column k=2 of A327803.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Sep 26 2019