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A009574 Expansion of e.g.f. sinh(log(1+x))*exp(x). 5
0, 1, 1, 3, -2, 25, -129, 931, -7412, 66753, -667475, 7342291, -88107414, 1145396473, -16035550517, 240533257875, -3848532125864, 65425046139841, -1177650830516967, 22375365779822563, -447507315596451050, 9397653627525472281, -206748379805560389929 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
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
Sequence in context: A090883 A100645 A132960 * A059422 A102056 A300955
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
Extended with signs by Olivier Gérard, Mar 15 1997
First Mathematica program replaced by Harvey P. Dale, Mar 23 2015
STATUS
approved

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Last modified May 23 07:28 EDT 2024. Contains 372760 sequences. (Running on oeis4.)