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

0,4

Links

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

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)

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 *)

Prog

(Maxima)
a(n):=n!/2*sum((k+2)*(-1)^(n-k+1)/k!, k, 0, n-1); /* Vladimir Kruchinin, Dec 30 2016 */
(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
(PARI) x='x+O('x^30); concat([0], Vec(serlaplace(sinh(log(1+x))*exp(x)))) \\ G. C. Greubel, Jan 21 2018
(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

Crossrefs

Cf. A000166, A000240, A054516, A103519.

Sequence in context: A090883 A100645 A132960 * A059422 A102056 A300955

Adjacent sequences: A009571 A009572 A009573 * A009575 A009576 A009577

Keyword

sign,easy

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

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

Status

approved