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

0, 0, 0, 3, 16, 125, 711, 5915, 46264, 438681, 4371085, 49321745, 588219523, 7751724513, 108240044745, 1633289839823, 26102966544024, 445098171557393, 8006283582196761, 152353662601600853, 3046062181913575921, 64015245150903376151, 1408108698825029286195

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=2 of A327803.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*
      multinomial(n, n-i*j, i$j), j=0..n/i))), x, 3)
    end:
a:= n-> coeff(b(n$2), x, 2):
seq(a(n), n=0..25);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
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}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 2}];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * n!, where c = Sum_{k>=2} 1/(k! - 1) = A331373 = 1.253498755699953471643360937905798940369232208332... - Vaclav Kotesovec, Sep 28 2019, updated Jul 19 2021