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A000795 Salié numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!.
(Formerly M2044 N0810)
18
1, 2, 12, 152, 3472, 126752, 6781632, 500231552, 48656756992, 6034272215552, 929327412759552, 174008703107274752, 38928735228629389312, 10255194381004799025152, 3142142941901073853366272, 1107912434323301224813002752, 445427836895850552387642130432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

L. 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

P. Bala, A triangle for calculating A000795.

L. Carlitz, The coefficients of cosh x/ cos x, Monatshefte für Mathematik 69(2) (1965), 129-135.

T. Chow and R. Stanley, Salié permutations and fair permutations

Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), #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, 31 (1958), 185-191.

J. M. Gandhi and V. S. Taneja, The coefficients of cosh x / cos x, Fib. Quart 10(4) (1972), 349-353.

M. S. Krick, On the coefficients of cosh x / cos x, Math. Mag., 34 (1960), 37-40.

Peter Luschny, An old operation on sequences: the Seidel transform.

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

EXAMPLE

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 + ...

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 *)

PROG

(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

CROSSREFS

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

Cf. A000364, A086646.

Sequence in context: A126345 A229558 A208582 * A085628 A177777 A053549

Adjacent sequences:  A000792 A000793 A000794 * A000796 A000797 A000798

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 18 14:52 EDT 2019. Contains 328161 sequences. (Running on oeis4.)