OFFSET
0,3
COMMENTS
There is another sequence called Salié numbers, A000795. - Benedict W. J. Irwin, Feb 10 2016
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 87, Problem 32.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
P. Bala, A triangle for calculating A005647
L. Carlitz, The coefficients of cosh x/ cos x, Monatshefte für Mathematik 69(2) (1965), 129-135.
FORMULA
a(n) = A000795(n)/2^n.
Expand cosh x / cos x and multiply coefficients by n!/(2^(n/2)).
a(n) = 2^(-n)*Sum_{k=0..n} A000364(k)*binomial(2*n, 2*k)). - Philippe Deléham, Jul 30 2003
a(n) ~ (2*n)! * 2^(n+2) * cosh(Pi/2) / Pi^(2*n+1). - Vaclav Kotesovec, Mar 08 2014
G.f.: A(x) = 1/(1 - x/(1 - 2x/(1 - 5x/(1 - 8x/(1 - 13x/(1 - 18x/(1 -...))))))), a continued fraction where the coefficients are A000982 (ceiling(n^2/2)). - Benedict W. J. Irwin, Feb 10 2016
MATHEMATICA
nmax = 17; se = Series[ Cosh[x]/Cos[x], {x, 0, 2*nmax}]; a[n_] := Coefficient[se, x, 2*n]*(2*n)!/2^n; Table[a[n], {n, 0, nmax}](* Jean-François Alcover, May 11 2012 *)
Join[{1}, Table[SeriesCoefficient[Series[1/(1+ContinuedFractionK[Floor[(k^2+ 1)/2]*x*-1, 1, {k, 1, 20}]), {x, 0, 20}], n], {n, 1, 20}]](* Benedict W. J. Irwin, Feb 10 2016 *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved