A344050 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*|Lah(n, k)|. Inverse binomial convolution of the unsigned Lah numbers A271703.

1, 1, -3, 1, 73, -699, 3001, 24697, -783999, 10946233, -80958779, -656003919, 40097528857, -944102982419, 14449693290033, -81180376526759, -4110744092532479, 203618771909117937, -5868277577182238579, 117997016943575159713, -1055340561026036009559, -45279878749358024400299

Offset: 0

Author

Peter Luschny, May 10 2021

Keywords

• sign

Crossrefs

Cf. A271703, A111596, A344050.

Programs

Maple

aList := proc(len) local lah;
lah := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1)*n!/k!):
seq(add((-1)^(n-k)*binomial(n, k)*lah(n, k), k = 0..n), n = 0..len-1) end:
print( aList(22) );

Mathematica

a[n_] := (-1)^(n-1) n n! HypergeometricPFQ[{1 - n, 1 - n}, {2, 2}, -1]; a[0] := 1;
Table[a[n], {n, 0, 20}]

Formula

a(n) = (-1)^(n-1)*n*n!*hypergeom([1 - n, 1 - n], [2, 2], -1) for n >= 1.