A009132 Expansion of e.g.f. cosh(log(1+x))/exp(x).

1, -1, 2, -7, 31, -161, 976, -6847, 54797, -493201, 4932046, -54252551, 651030667, -8463398737, 118487582396, -1777313736031, 28437019776601, -483429336202337, 8701728051642202, -165332832981201991, 3306656659624039991

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A009578.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cosh(Log(1+x))/Exp(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 28 2018

Mathematica

With[{nmax = 30}, CoefficientList[Series[Cosh[Log[1 + x]]/Exp[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jul 28 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace(cosh(log(1+x))/exp(x))) \\ G. C. Greubel, Jul 28 2018

Formula

a(n) = (-1)^n/2*floor(n!*exp(1)-n+1), n>0. - Vladeta Jovovic, Aug 17 2002

E.g.f.: (2+2*x+x^2)/(2*(1+x)*exp(x)) = (1+x/G(0))/2 ; G(k) = 1 - 1/(1 + x + x^2 - (x*(1+x))^2/(x*(1+x) - (k+1)/(1 + x/G(k+1)))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 28 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997