A000795 Salié numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!.

1, 2, 12, 152, 3472, 126752, 6781632, 500231552, 48656756992, 6034272215552, 929327412759552, 174008703107274752, 38928735228629389312, 10255194381004799025152, 3142142941901073853366272, 1107912434323301224813002752, 445427836895850552387642130432

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Named after the German mathematician Hans Salié (1902-1978). - Amiram Eldar, Jun 10 2021

Examples

			cosh x / cos x = Sum_{n>=0} a(n)*x^(2n)/(2n)! = 1 + x^2 + (1/2)*x^4 + (19/90)*x^6 + (31/360)*x^8 + (3961/113400)*x^10 + ...
G.f. = 1 + 2*x + 12*x^2 + 252*x^3 + 3472*x^4 + 126752*x^5 + 6781632*x^6 + ...
		

References

• Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 86, Problem 32.
• Hans Salié, Arithmetische Eigenschaften der Koeffizienten einer speziellen Hurwitzschen Potenzreihe, Wiss. Z. Karl-Marx-Univ. Leipzig Math.-Natur. Reihe 12 (1963), pp. 617-618.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• 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
• Peter Bala, A triangle for calculating A000795, 2017.
• L. Carlitz, The coefficients of cosh x/ cos x, Monatshefte für Mathematik, Vol. 69, No. 2 (1965), pp. 129-135.
• Timothy Chow and Richard Stanley, Salié permutations and fair permutations, MathOverflow, 2012.
• Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.
• J. M. Gandhi, The coefficients of cosh x/ cos x and a note on Carlitz's coefficients of sinh x / sin x, Math. Magazine, Vol. 31, No. 4 (1958), pp. 185-191..
• J. M. Gandhi and V. S. Taneja, The coefficients of cosh x / cos x, Fib. Quart., Vol. 10, No. 4 (1972), pp. 349-353.
• M. S. Krick, On the coefficients of cosh x / cos x, Math. Mag., Vol. 34, No. 1 (1960), pp. 37-40.
• Peter Luschny, An old operation on sequences: the Seidel transform.

Crossrefs

A005647(n) = a(n)/2^n.

Cf. A000364, A086646.

Programs

Maple

A000795 := proc(n)
        (2*n)!*coeftayl( cosh(x)/cos(x),x=0,2*n) ;
end proc: # R. J. Mathar, Oct 20 2011

Mathematica

max = 16; se = Series[ Cosh[x] / Cos[x], {x, 0, 2*max} ]; a[n_] := SeriesCoefficient[ se, 2*n ]*(2*n)!; Table[ a[n], {n, 0, max} ] (* Jean-François Alcover, Apr 02 2012 *)
With[{nn=40},Take[CoefficientList[Series[Cosh[x]/Cos[x],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, May 11 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ Cosh[ x] / Cos[ x], {x, 0, m}]]]; (* Michael Somos, Aug 15 2015 *)

Sage

# Generalized algorithm of L. Seidel (1877)
def A000795_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1 if e == 1 else 0
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        if e == -1 : R.append(A[-i//2])
    return R
A000795_list(10) # Peter Luschny, Jun 02 2012

Formula

a(n) = Sum_{k=0..n} binomial(2n, 2k)*A000364(n-k). - Philippe Deléham, Dec 16 2003

a(n) = Sum_{k>=0} (-1)^(n+k)*2^(2n-k)*A065547(n, k). - Philippe Deléham, Feb 26 2004

a(n) = Sum_{k>=0} A086646(n, k). - Philippe Deléham, Feb 26 2004

G.f.: 1 / (1 - (1^2+1)*x / (1 - 2^2*x / (1 - (3^2+1)*x / (1 - 4^2*x / (1 - (5^2+1)*x / (1 - 6^2*x / ...)))))). - Michael Somos, May 12 2012

G.f.: Q(0)/(1-2*x), where Q(k) = 1 - 8*x^2*(2*k^2+2*k+1)*(k+1)^2/( 8*x^2*(2*k^2+2*k+1)*(k+1)^2 - (1 - 8*x*k^2 - 4*x*k -2*x)*(1 - 8*x*k^2 - 20*x*k -14*x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013

a(n) ~ (2*n)! * 2^(2*n+2) * cosh(Pi/2) / Pi^(2*n+1). - Vaclav Kotesovec, Mar 08 2014

a(n) = 1 - Sum_{k=1..n} (-1)^k * binomial(2*n,2*k) * a(n-k). - Ilya Gutkovskiy, Mar 09 2022