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

%I A000795 M2044 N0810 #86 Jan 05 2025 19:51:31
%S A000795 1,2,12,152,3472,126752,6781632,500231552,48656756992,6034272215552,
%T A000795 929327412759552,174008703107274752,38928735228629389312,
%U A000795 10255194381004799025152,3142142941901073853366272,1107912434323301224813002752,445427836895850552387642130432
%N A000795 Salié numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!.
%C A000795 Named after the German mathematician Hans Salié (1902-1978). - _Amiram Eldar_, Jun 10 2021
%D A000795 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 86, Problem 32.
%D A000795 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 A000795 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000795 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000795 T. D. Noe, <a href="/A000795/b000795.txt">Table of n, a(n) for n = 0..100</a>
%H A000795 Peter Bala, <a href="/A000795/a000795.pdf">A triangle for calculating A000795</a>, 2017.
%H A000795 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 A000795 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 A000795 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 A000795 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 A000795 J. M. Gandhi and V. S. Taneja, <a href="https://web.archive.org/web/2024*/https://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 A000795 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 A000795 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%F A000795 a(n) = Sum_{k=0..n} binomial(2n, 2k)*A000364(n-k). - _Philippe Deléham_, Dec 16 2003
%F A000795 a(n) = Sum_{k>=0} (-1)^(n+k)*2^(2n-k)*A065547(n, k). - _Philippe Deléham_, Feb 26 2004
%F A000795 a(n) = Sum_{k>=0} A086646(n, k). - _Philippe Deléham_, Feb 26 2004
%F A000795 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 A000795 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 A000795 a(n) ~ (2*n)! * 2^(2*n+2) * cosh(Pi/2) / Pi^(2*n+1). - _Vaclav Kotesovec_, Mar 08 2014
%F A000795 a(n) = 1 - Sum_{k=1..n} (-1)^k * binomial(2*n,2*k) * a(n-k). - _Ilya Gutkovskiy_, Mar 09 2022
%e A000795 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 A000795 G.f. = 1 + 2*x + 12*x^2 + 252*x^3 + 3472*x^4 + 126752*x^5 + 6781632*x^6 + ...
%p A000795 A000795 := proc(n)
%p A000795         (2*n)!*coeftayl( cosh(x)/cos(x),x=0,2*n) ;
%p A000795 end proc: # _R. J. Mathar_, Oct 20 2011
%t A000795 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 A000795 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 A000795 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 A000795 (Sage) # Generalized algorithm of L. Seidel (1877)
%o A000795 def A000795_list(n) :
%o A000795     R = []; A = {-1:0, 0:0}
%o A000795     k = 0; e = 1
%o A000795     for i in range(n) :
%o A000795         Am = 1 if e == 1 else 0
%o A000795         A[k + e] = 0
%o A000795         e = -e
%o A000795         for j in (0..i) :
%o A000795             Am += A[k]
%o A000795             A[k] = Am
%o A000795             k += e
%o A000795         if e == -1 : R.append(A[-i//2])
%o A000795     return R
%o A000795 A000795_list(10) # _Peter Luschny_, Jun 02 2012
%Y A000795 A005647(n) = a(n)/2^n.
%Y A000795 Cf. A000364, A086646.
%K A000795 nonn,easy,nice
%O A000795 0,2
%A A000795 _N. J. A. Sloane_