A009454 Expansion of e.g.f. sin(log(1+x)).

0, 1, -1, 1, 0, -10, 90, -730, 6160, -55900, 549900, -5864300, 67610400, -839594600, 11186357000, -159300557000, 2416003824000, -38894192662000, 662595375078000, -11911522255750000, 225382826562400000

Offset

0,6

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Sum_{k=0..n-1} (-1)^k*T(n-1, k)*cos(Pi*(n-k-1)/2); T(n, k) = abs(A008276(n, k)). - Paul Barry, Apr 18 2005

abs(a(n)) = abs(Re(Product_{k=1..n-1} (k+I))) with I^2 = -1. - Yalcin Aktar, Jul 02 2005

a(n+2) = -(2n+1)*a(n+1)-(n^2+1)*a(n), a(0)=0, a(1)=1. - Remy Lachaud (pacifik31(AT)aol.com), Dec 25 2005

a(n) = Sum_{k=0..n/2} Stirling1(n,2k+1)*(-1)^k. - Vladimir Kruchinin, Aug 03 2010

Mathematica

CoefficientList[Series[Sin[Log[1+x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2015 *)
FullSimplify[Table[-((-1)^n*(Gamma[1 + I]*Gamma[-I + n] + Gamma[1 - I]*Gamma[I + n])*Sinh[Pi]) / (2*Pi), {n, 0, 20}]] (* Vaclav Kotesovec, Jan 24 2015 *)
Table[-(-1)^n Re[Pochhammer[1+I, n-1]], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 13 2016 *)

Prog

(Maxima) sum(stirling1(n, 2*k+1)*(-1)^(k), k, 0, n/2) /* Vladimir Kruchinin, Aug 03 2010 */
(Python)
from sympy.functions.combinatorial.numbers import stirling
def A009454(n): return sum(stirling(n, (k<<1)+1, kind=1, signed=True)*(-1 if k&1 else 1) for k in range(n+1>>1)) # Chai Wah Wu, Feb 22 2024

Crossrefs

Cf. A242652, A231530.

Sequence in context: A159733 A265325 A038726 * A231530 A242652 A291392

Adjacent sequences: A009451 A009452 A009453 * A009455 A009456 A009457

Keyword

sign,easy

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Status

approved