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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 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

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