A000795 - OEIS

%I M2044 N0810 #84 Mar 10 2022 03:24:20

%S 1,2,12,152,3472,126752,6781632,500231552,48656756992,6034272215552,

%T 929327412759552,174008703107274752,38928735228629389312,

%U 10255194381004799025152,3142142941901073853366272,1107912434323301224813002752,445427836895850552387642130432

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

%C Named after the German mathematician Hans Salié (1902-1978). - _Amiram Eldar_, Jun 10 2021

%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 86, Problem 32.

%D Hans Salié, Arithmetische Eigenschaften der Koeffizienten einer speziellen Hurwitzschen Potenzreihe, Wiss. Z. Karl-Marx-Univ. Leipzig Math.-Natur. Reihe 12 (1963), pp. 617-618.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000795/b000795.txt">Table of n, a(n) for n = 0..100</a>

%H Peter Bala, <a href="/A000795/a000795.pdf">A triangle for calculating A000795</a>, 2017.

%H L. Carlitz, <a href="https://doi.org/10.1007/BF01298317">The coefficients of cosh x/ cos x</a>, Monatshefte für Mathematik, Vol. 69, No. 2 (1965), pp. 129-135.

%H Timothy Chow and Richard Stanley, <a href="http://mathoverflow.net/questions/113983/sali%C3%A9-permutations-and-fair-permutations">Salié permutations and fair permutations</a>, MathOverflow, 2012.

%H Marc Deléglise and Jean-Louis Nicolas, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Deleglise/deleglise3.html">On the Largest Product of Primes with Bounded Sum</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.

%H J. M. Gandhi, <a href="http://www.jstor.org/stable/3029194">The coefficients of cosh x/ cos x and a note on Carlitz's coefficients of sinh x / sin x</a>, Math. Magazine, Vol. 31, No. 4 (1958), pp. 185-191..

%H J. M. Gandhi and V. S. Taneja, <a href="http://www.fq.math.ca/Scanned/10-4/gandhi.pdf">The coefficients of cosh x / cos x</a>, Fib. Quart., Vol. 10, No. 4 (1972), pp. 349-353.

%H M. S. Krick, <a href="http://www.jstor.org/stable/2687850">On the coefficients of cosh x / cos x</a>, Math. Mag., Vol. 34, No. 1 (1960), pp. 37-40.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.

%F a(n) = Sum_{k=0..n} binomial(2n, 2k)*A000364(n-k). - _Philippe Deléham_, Dec 16 2003

%F a(n) = Sum_{k>=0} (-1)^(n+k)*2^(2n-k)*A065547(n, k). - _Philippe Deléham_, Feb 26 2004

%F a(n) = Sum_{k>=0} A086646(n, k). - _Philippe Deléham_, Feb 26 2004

%F 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

%F 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

%F a(n) ~ (2*n)! * 2^(2*n+2) * cosh(Pi/2) / Pi^(2*n+1). - _Vaclav Kotesovec_, Mar 08 2014

%F a(n) = 1 - Sum_{k=1..n} (-1)^k * binomial(2*n,2*k) * a(n-k). - _Ilya Gutkovskiy_, Mar 09 2022

%e 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 + ...

%e G.f. = 1 + 2*x + 12*x^2 + 252*x^3 + 3472*x^4 + 126752*x^5 + 6781632*x^6 + ...

%p A000795 := proc(n)

%p         (2*n)!*coeftayl( cosh(x)/cos(x),x=0,2*n) ;

%p end proc: # _R. J. Mathar_, Oct 20 2011

%t 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 *)

%t 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 *)

%t a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ Cosh[ x] / Cos[ x], {x, 0, m}]]]; (* _Michael Somos_, Aug 15 2015 *)

%o (Sage) # Generalized algorithm of L. Seidel (1877)

%o def A000795_list(n) :

%o     R = []; A = {-1:0, 0:0}

%o     k = 0; e = 1

%o     for i in range(n) :

%o         Am = 1 if e == 1 else 0

%o         A[k + e] = 0

%o         e = -e

%o         for j in (0..i) :

%o             Am += A[k]

%o             A[k] = Am

%o             k += e

%o         if e == -1 : R.append(A[-i//2])

%o     return R

%o A000795_list(10) # _Peter Luschny_, Jun 02 2012

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

%Y Cf. A000364, A086646.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_