OFFSET
0,2
COMMENTS
Named after the German mathematician Hans Salié (1902-1978). - Amiram Eldar, Jun 10 2021
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.
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
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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved