A009574 Expansion of e.g.f. sinh(log(1+x))*exp(x).

0, 1, 1, 3, -2, 25, -129, 931, -7412, 66753, -667475, 7342291, -88107414, 1145396473, -16035550517, 240533257875, -3848532125864, 65425046139841, -1177650830516967, 22375365779822563, -447507315596451050, 9397653627525472281, -206748379805560389929

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A000166, A000240, A054516, A103519.

Programs

Magma

[0] cat [(&+[(k+2)*(-1)^(n-k+1)/Factorial(k): k in [0..n-1]])*( Factorial(n)/2): n in [1..30]]; // G. C. Greubel, Jan 21 2018

Maple

seq(n*(1-(-1)^n*A000166(n-1))/2,n=0..20); # Peter Luschny, Dec 30 2016

Mathematica

CoefficientList[Series[(E^x*x*(2 + x))/(2*(1 + x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2015 *)
With[{nn=20},CoefficientList[Series[Sinh[Log[1+x]]*Exp[x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 23 2015 *)
Table[(-1)^n*n*((-1)^n-Subfactorial[n-1])/2,{n,0,20}] (* Peter Luschny, Dec 30 2016 *)

Maxima

a(n):=n!/2*sum((k+2)*(-1)^(n-k+1)/k!,k,0,n-1); /* Vladimir Kruchinin, Dec 30 2016 */

PARI

x='x+O('x^30); concat([0], Vec(serlaplace(sinh(log(1+x))*exp(x)))) \\ G. C. Greubel, Jan 21 2018

Sage

def A009574():
    a, n = 0, 0
    while True:
        yield a//2
        n += 1
        a = n*(n+1-a)
a = A009574(); [next(a) for  in (0..20)] # _Peter Luschny, Dec 30 2016

Formula

a(n) ~ n! * (-1)^(n+1) / (2*exp(1)). - Vaclav Kotesovec, Jan 23 2015

a(n) = n!/2*Sum_{k=0..n-1}(k+2)*(-1)^(n-k+1)/k!. - Vladimir Kruchinin, Dec 30 2016

a(n) = n*(1-(-1)^n*SF(n-1))/2, where SF(n) is the subfactorial A000166. - Peter Luschny, Dec 30 2016

From Seiichi Manyama, Dec 31 2023: (Start)

a(0) = 0; a(n) = -n*a(n-1) + binomial(n+1,2).

E.g.f.: x * (1+x/2) * exp(x) / (1+x). (End)

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

First Mathematica program replaced by Harvey P. Dale, Mar 23 2015