A005647 Salié numbers.

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

Offset: 0

Author

Simon Plouffe, N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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.

Crossrefs

Cf. A000795, A000982.

Programs

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 *)

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