A009628 Expansion of e.g.f.: sinh(x)/(1+x).

0, 1, -2, 7, -28, 141, -846, 5923, -47384, 426457, -4264570, 46910271, -562923252, 7318002277, -102452031878, 1536780478171, -24588487650736, 418004290062513, -7524077221125234, 142957467201379447, -2859149344027588940, 60042136224579367741

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Comments

(-1)^n*(A000166 + A000522)/2 = A009179, (-1)^n*(A000166-A000522)/2 = this_sequence.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A000166, A000522, A009179, A051397, A087208, A186763.

Programs

Maple

G(x):= sinh(x)/(1+x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009

Mathematica

a[n_] := (-1)^n (Exp[-1] Gamma[1 + n, -1] - Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 18 2017 *)
With[{nn=30},CoefficientList[Series[Sinh[x]/(1+x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 19 2023 *)

PARI

a(n) = n!*polcoeff((sinh(x)/(1+x) + x * O(x^n)), n) \\ Charles R Greathouse IV, Sep 09 2016

PARI

x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)/(1+x)))) \\ Altug Alkan, Dec 18 2017

Ruby

def A009628(n)
  a = 0
  (0..n).map{|i| a = -i * a + i % 2}
end # Seiichi Manyama, Sep 09 2016

Formula

a(n) = (-1)^(n+1)*floor(n!*sinh(1)), n>=1. - Vladeta Jovovic, Aug 10 2002

Let u(1) = 1, u(n) = n*u(n-1) + n (mod 2); then for n>0, a(n) = (-1)^(n+1)*u(n). - Benoit Cloitre, Jan 12 2003

Unsigned sequence satisfies a(n) = n*a(n-1)+a(n-2)-(n-2)*a(n-3), with E.g.f. sinh(z)/(1-z). - Mario Catalani (mario.catalani(AT)unito.it), Feb 08 2003

a(n) = (-1)^(n+1) * n! * Sum_{k=1..floor((n+1)/2)} 1/(2*k-1)!.

a(n) = -n*a(n-1) + n (mod 2). - Seiichi Manyama, Sep 09 2016

a(n) = (-1)^n*(exp(-1)*Gamma(1+n,-1) - exp(1)*Gamma(1+n,1))/2. - Peter Luschny, Dec 18 2017

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified by Harvey P. Dale, Mar 19 2023