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A327826 Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size two.

%I A327826 #15 Mar 18 2024 06:01:40
%S A327826 0,0,0,3,16,125,711,5915,46264,438681,4371085,49321745,588219523,
%T A327826 7751724513,108240044745,1633289839823,26102966544024,445098171557393,
%U A327826 8006283582196761,152353662601600853,3046062181913575921,64015245150903376151,1408108698825029286195
%N A327826 Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size two.
%H A327826 Alois P. Heinz, <a href="/A327826/b327826.txt">Table of n, a(n) for n = 0..450</a>
%H A327826 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A327826 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327826 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 A327826 with(combinat):
%p A327826 b:= proc(n, i) option remember; series(`if`(n=0, 1,
%p A327826      `if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*
%p A327826       multinomial(n, n-i*j, i$j), j=0..n/i))), x, 3)
%p A327826     end:
%p A327826 a:= n-> coeff(b(n$2), x, 2):
%p A327826 seq(a(n), n=0..25);
%t A327826 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A327826 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 A327826 a[n_] := SeriesCoefficient[b[n, n], {x, 0, 2}];
%t A327826 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A327826 Column k=2 of A327803.
%K A327826 nonn
%O A327826 0,4
%A A327826 _Alois P. Heinz_, Sep 26 2019