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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Wikipedia, Multinomial coefficients

Wikipedia, Partition (number theory)

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

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 *)

Crossrefs

Column k=2 of A327803.

Sequence in context: A090135 A351423 A188417 * A157457 A365626 A000950

Adjacent sequences: A327823 A327824 A327825 * A327827 A327828 A327829

Keyword

nonn

Author

Alois P. Heinz, Sep 26 2019

Status

approved