login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000795 Salié numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!.
(Formerly M2044 N0810)
13
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.

M. Deléglise, J.-L. Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, 18 (2015), #15.2.8.

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

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

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, V. S. Taneja, The coefficients of cosh x / cos x, Fib. Quart 10 (4) (1972) 349

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, C(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..inf} 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 20 06:45 EST 2018. Contains 299358 sequences. (Running on oeis4.)