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URL: https://oeis.org/A003728

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A003728
Expansion of e.g.f. log(1+x*cos(x)).
2
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
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
KEYWORD
sign
EXTENSIONS
Corrected title, Joerg Arndt, Apr 23 2011
STATUS
approved