A003728 Expansion of e.g.f. log(1+x*cos(x)). (Formerly M4208)

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

Offset

0,5

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.

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

Mathematica

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

Prog

(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

Crossrefs

Sequence in context: A365301 A337574 A166786 * A216370 A225425 A267890

Adjacent sequences: A003725 A003726 A003727 * A003729 A003730 A003731

Keyword

sign

Author

R. H. Hardin, Simon Plouffe

Extensions

Corrected title, Joerg Arndt, Apr 23 2011

Status

approved