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A003703 Expansion of e.g.f. cos(log(1+x)).
(Formerly M2856)
16
1, 0, -1, 3, -10, 40, -190, 1050, -6620, 46800, -365300, 3103100, -28269800, 271627200, -2691559000, 26495469000, -238131478000, 1394099824000, 15194495654000, -936096296850000, 29697351895900000, -819329864480400000, 21683886333440500000, -570263312237604700000 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A258973 A217885 A216367 * A242651 A231531 A136128
KEYWORD
easy,sign
AUTHOR
STATUS
approved

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Last modified April 25 09:38 EDT 2024. Contains 371967 sequences. (Running on oeis4.)