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)=Product_{k=1..n} i+k (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
a(n) = Re(gamma(i+1)/gamma(i+1-n)). The imaginary part is A009454. - Colin Beveridge, Jul 30 2024
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
KEYWORD
easy,sign
AUTHOR
STATUS
approved