A003728 Expansion of e.g.f. log(1+x*cos(x)).

0, 1, -1, -1, 6, -31, 120, -337, -784, 24705, -288000, 2451679, -14032128, -17936543, 2173889536, -42895630065, 583266662400, -5396647099903, 5119183650816, 1239561882325439, -36754121131294720, 708575518706816481

Offset: 0

Author

R. H. Hardin, Simon Plouffe

Keywords

• sign

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Log[1+Cos[x]x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 11 2011 *)

Maxima

a(n) := n! *sum(((sum((k-2*i)^(n-k)*binomial(k,i),i,0,floor((k-1)/2)))*(-1)^((n-k)/2)*((-1)^(n-k)+1))/(2^k*(n-k)!)/k*(-1)^(k-1),k,1,n-1)+(-1)^(n-1)*(n-1)!; /* Vladimir Kruchinin, Apr 23 2011 */

PARI

my(x='x+O('x^30)); concat(0, Vec(serlaplace(log(1+x*cos(x))))) \\ Michel Marcus, Oct 29 2022

Formula

a(0)=0 and for n>=1, a(n)n!*sum(k=1..n-1,((sum(i=0,floor((k-1)/2),(k-2*i)^(n-k)*binomial(k,i)))*(-1)^((n-k)/2)*((-1)^(n-k)+1))/(2^k*(n-k)!)/k*(-1)^(k-1))+(-1)^(n-1)*(n-1)!. - Vladimir Kruchinin, Apr 23 2011

Extensions

Corrected title, Joerg Arndt, Apr 23 2011