A009179 E.g.f. cosh(x)/(1+x).

1, -1, 3, -9, 37, -185, 1111, -7777, 62217, -559953, 5599531, -61594841, 739138093, -9608795209, 134523132927, -2017846993905, 32285551902481, -548854382342177, 9879378882159187, -187708198761024553, 3754163975220491061

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Comments

Unsigned sequence satisfies a(n)=n*a(n-1)+a(n-2)-(n-2)*a(n-3), a(0)=1,a(1)=1,a(2)=3 with e.g.f. cosh(z)/(1-z). - Mario Catalani (mario.catalani(AT)unito.it), Feb 07 2003

(-1)^n*(A000166(n) + A000522(n))/2 = this_sequence, (-1)^n*(A000166(n) - A000522(n))/2 = A009628(n).

The positive sequence has e.g.f. cosh(x)/(1-x), with a(n)=sum{k=0..floor(n/2), binomial(n,2k)(n-2k)!}. It is the mean of the binomial and inverse binomial transforms of n!. - Paul Barry, May 01 2005

Links

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

Crossrefs

Cf. A000166, A000522, A001540, A009628.

Programs

Maple

restart: G(x):= cosh(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 *)

PARI

x='x+O('x^99); Vec(serlaplace(cosh(x)/(1+x))) \\ Altug Alkan, Dec 18 2017

Formula

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

a(n) = (1+(-1)^n)/2-n*a(n-1). - Vladeta Jovovic, Apr 19 2003

a(n) = (-1)^n * n! * sum{k=0, [n/2], 1/(2k)!}.

E.g.f.: U(0)/(1+x) where U(k)= 1 + x^2/((4*k+1)*(4*k+2) - x^2*(4*k+1)*(4*k+2)/(x^2 + (4*k+3)*(4*k+4)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2012

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