%I M4208 #40 Oct 29 2022 09:37:49
%S 0,1,-1,-1,6,-31,120,-337,-784,24705,-288000,2451679,-14032128,
%T -17936543,2173889536,-42895630065,583266662400,-5396647099903,
%U 5119183650816,1239561882325439,-36754121131294720,708575518706816481
%N Expansion of e.g.f. log(1+x*cos(x)).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A003728/b003728.txt">Table of n, a(n) for n = 0..200</a>
%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F 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
%t With[{nn=30},CoefficientList[Series[Log[1+Cos[x]x],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 11 2011 *)
%o (Maxima)
%o 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 */
%o (PARI) my(x='x+O('x^30)); concat(0, Vec(serlaplace(log(1+x*cos(x))))) \\ _Michel Marcus_, Oct 29 2022
%K sign
%O 0,5
%A _R. H. Hardin_, _Simon Plouffe_
%E Corrected title, _Joerg Arndt_, Apr 23 2011