A260845 a(n) = Sum_{k=0..n} (-1)^k*P(n,k)*k!, where P(n,k) is the number of partitions of n into k parts.

1, -1, 1, -5, 21, -105, 635, -4507, 36457, -330971, 3334377, -36913947, 445426739, -5818545721, 81805507069, -1231690773053, 19772941871385, -337146625794753, 6085005877228943, -115897323408009187, 2323090928155541677, -48883768421712917555, 1077440388662366900397

Offset: 0

Author

Peter Luschny, Aug 01 2015

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A008284, A101880.

Row sums of A318144.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1,
      b(n, i-1), 0)+expand(b(n-i, min(n-i, i))*x))
    end:
a:= n-> (p-> add(i!*coeff(p, x, i)*(-1)^i, i=0..n))(b(n$2)):
seq(a(n), n=0..27);  # Alois P. Heinz, Sep 18 2019

Mathematica

CoefficientList[ Series[ Sum[ n!(-x)^n / Product[1 - x^k, {k, n}], {n, 0, 22}], {x, 0, 22}], x]

Sage

from sage.combinat.partition import number_of_partitions_length
[sum([(-1)^k*number_of_partitions_length(n,k)*factorial(k) for k in (0..n)]) for n in (0..22)]

Formula

G.f.: Sum(n!*(-x)^n/Product(1-x^k, k=1..n), n=1..infinity).