A003703 Expansion of e.g.f. cos(log(1+x)). (Formerly M2856)

1, 0, -1, 3, -10, 40, -190, 1050, -6620, 46800, -365300, 3103100, -28269800, 271627200, -2691559000, 26495469000, -238131478000, 1394099824000, 15194495654000, -936096296850000, 29697351895900000, -819329864480400000, 21683886333440500000, -570263312237604700000

Offset

0,4

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (first 100 terms from T. D. Noe)

Vaclav Kotesovec, Graph a(n+1)/a(n)

Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Formula

a(n) = sum{k=0..n-1, (-1)^(k+1)*T(n-k, k)*sin(pi*(n-k-1)/2)}+0^n; T(n, k)=abs(A008276(n, k)). - Paul Barry, Apr 18 2005

abs(a(n)) = abs(f(n)) with f(n)=prod(i+k,k=1..n) (where i^2=-1). - Yalcin Aktar, Jul 13 2009

a(n) = Sum_{k=0..floor(n/2)} stirling1(n,2*k)*(-1)^(k). - Vladimir Kruchinin, Jan 29 2011

a(n+2)= -a(n+1)*(2*n+1) - a(n)*(1+n^2), a(0)=1, a(1)=0. - Sergei N. Gladkovskii, Aug 17 2012

a(n) = (-1)^n * ( (i)_n + (-i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit. - Seiichi Manyama, Oct 10 2022

Example

1 - x^2 + 3*x^3 - 10*x^4 + 40*x^5 - 190*x^6 + 1050*x^7 - 6620*x^8 + ...

Maple

a:= n-> add(Stirling1(n, 2*k) * (-1)^(k), k=0..floor(n/2)):
seq(a(n), n=0..20);

Mathematica

CoefficientList[Series[Cos[Log[1 + x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 16 2015 *)
Table[(-1)^n Im[Pochhammer[1-I, n-1]], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 13 2016 *)

Prog

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( cos( log( 1 + x + x * O(x^n))), n))} /* Michael Somos, Jul 26 2012 */
(PARI) a(n) = (-1)^n*(prod(k=0, n-1, I+k)+prod(k=0, n-1, -I+k))/2; \\ Seiichi Manyama, Oct 10 2022
(Python)
from sympy.functions.combinatorial.numbers import stirling
def A003703(n): return sum(stirling(n, k<<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. A009024, A009454.

Sequence in context: A258973 A217885 A216367 * A242651 A231531 A136128

Adjacent sequences: A003700 A003701 A003702 * A003704 A003705 A003706

Keyword

easy,sign

Author

R. H. Hardin, Simon Plouffe

Status

approved