A120765 Expansion of e.g.f. -exp(-x)*log(1-2*x)/2.

0, 1, 0, 5, 24, 209, 2120, 25829, 365456, 5895105, 106794992, 2147006949, 47436635752, 1142570789073, 29797622256376, 836527783016197, 25153234375160992, 806519154686509057, 27470342073410272608, 990496662138073867333, 37692249497898323450424

Offset: 0

Author

Max Alekseyev, Jul 03 2006

Keywords

• nonn

Comments

Inverse binomial transform of even double factorials (A000165) with 0 prepended: [0, 1, 2, 8, 48 ...].

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013

Crossrefs

Cf. A000165, A000354.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, n*(2-n),
      (2*n-4)*(a(n-1) +a(n-3)) +(4*n-7)*a(n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 08 2013
A120765 := n -> (-1)^(n+1)*n*hypergeom([1-n,1,1],[2],2):
seq(simplify(A120765(n)), n=0..20); # Peter Luschny, May 09 2017

Mathematica

CoefficientList[Series[-E^(-x)*Log[1-2*x]/2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

PARI

my(x='x+O('x^33)); concat([0],Vec(serlaplace(-exp(-x)*log(1-2*x)/2))) \\ Joerg Arndt, Jun 29 2015

PARI

vector(30, n, n--; sum(k=0, n-1, (-1)^(n-1-k) * binomial(n,k+1) * k! * 2^k)) \\ Altug Alkan, Oct 28 2015

Formula

E.g.f.: -e^(-x)*log(1-2*x)/2.

a(n) = Sum_{i=0..n-1} (-1)^(n-1-i) * C(n,i+1) * i! * 2^i.

a(n) ~ (n-1)! * 2^(n-1) * exp(-1/2). - Vaclav Kotesovec, Oct 08 2013

From Vladimir Reshetnikov, Oct 28 2015: (Start)

a(n) = Sum_{k=0..n-1} A000354(k)*(-1)^(n+k+1).

Recurrence: a(0) = 0, a(1) = 1, a(2) = 0, a(n) = 2*(n-2)*a(n-3) + (4*n-7)*a(n-2) + 2*(n-2)*a(n-1). (End)

a(n) = (-1)^(n+1)*n*hypergeom([1-n,1,1],[2],2). - Peter Luschny, May 09 2017

Extensions

Formula corrected by Max Alekseyev, Sep 15 2009