A182927 Row sums of A182928.

1, 0, 3, -8, 25, -99, 721, -5704, 40881, -340325, 3628801, -41245511, 479001601, -6129725315, 87212177053, -1317906346184, 20922789888001, -354320889234597, 6402373705728001, -121882630320799633, 2432928081076384321, -51041048673495232715

Offset: 1

Author

Peter Luschny, Apr 16 2011

Keywords

• sign

Comments

The number of partitions of an n-set with distinct block sizes can

be computed recursively as A007837(0) = 1 and A007837(n) = - Sum_{1<=k<=n} binomial(n-1,k-1) * A182927(k) * A007837(n-k).

Möbius inversion yields: 1, -1, 2, -8, 24, -101, 720, -5696, 40878,...

A182927(2*i+1) = A182926(2*i+1)

Examples

			a(6) = 1 - 10 + 30 - 120 = -99.
		

Crossrefs

Cf. A182926, A182928, A005651, A007837.

Programs

Maple

A182927 := proc(n) local d;
add(-n! / (d*(-(n/d)!)^d), d = numtheory[divisors](n)) end:
seq(A182927(i), i = 1..22);

Mathematica

a[n_] := Sum[ -n!/(d*(-(n/d)!)^d), {d, Divisors[n]}]; Table[a[n], {n, 1, 22}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Formula

a(n) = Sum_{d|n} -n!/(d*(-(n/d)!)^d).

E.g.f.: Sum_{k>=1} log(1 + x^k/k!). - Ilya Gutkovskiy, May 21 2019