A005647 Salié numbers. (Formerly M3066)

1, 1, 3, 19, 217, 3961, 105963, 3908059, 190065457, 11785687921, 907546301523, 84965187064099, 9504085749177097, 1251854782837499881, 191781185418766714683, 33810804270120276636139, 6796689405759438360407137, 1545327493049348356667631841

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 A005467

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

Cf. A000795, A000982.

Sequence in context: A074707 A230317 A135749 * A158876 A001833 A001035

Adjacent sequences: A005644 A005645 A005646 * A005648 A005649 A005650

Keyword

nonn,easy,nice

Author

Simon Plouffe, N. J. A. Sloane

Status

approved