cp's OEIS Frontend

A005647 Salié numbers.

%I A005647 M3066 #46 Dec 28 2024 23:20:05
%S A005647 1,1,3,19,217,3961,105963,3908059,190065457,11785687921,907546301523,
%T A005647 84965187064099,9504085749177097,1251854782837499881,
%U A005647 191781185418766714683,33810804270120276636139,6796689405759438360407137,1545327493049348356667631841
%N A005647 Salié numbers.
%C A005647 There is another sequence called Salié numbers, A000795. - _Benedict W. J. Irwin_, Feb 10 2016
%D A005647 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 87, Problem 32.
%D A005647 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005647 T. D. Noe, <a href="/A005647/b005647.txt">Table of n, a(n) for n = 0..100</a>
%H A005647 P. Bala, <a href="/A005647/a005647.pdf">A triangle for calculating A005647</a>
%H A005647 L. Carlitz, <a href="https://doi.org/10.1007/BF01298317">The coefficients of cosh x/ cos x</a>, Monatshefte für Mathematik 69(2) (1965), 129-135.
%F A005647 a(n) = A000795(n)/2^n.
%F A005647 Expand cosh x / cos x and multiply coefficients by n!/(2^(n/2)).
%F A005647 a(n) = 2^(-n)*Sum_{k=0..n} A000364(k)*binomial(2*n, 2*k). - _Philippe Deléham_, Jul 30 2003
%F A005647 a(n) ~ (2*n)! * 2^(n+2) * cosh(Pi/2) / Pi^(2*n+1). - _Vaclav Kotesovec_, Mar 08 2014
%F A005647 G.f.: A(x) = 1/(1 - x/(1 - 2x/(1 - 5x/(1 - 8x/(1 - 13x/(1 - 18x/(1 -...))))))), a continued fraction where the coefficients are A000982 (ceiling(n^2/2)). - _Benedict W. J. Irwin_, Feb 10 2016
%t A005647 nmax = 17; se = Series[ Cosh[x]/Cos[x], {x, 0, 2*nmax}]; a[n_] := Coefficient[se, x, 2*n]*(2*n)!/2^n; Table[a[n], {n, 0, nmax}](* _Jean-François Alcover_, May 11 2012 *)
%t A005647 Join[{1},Table[SeriesCoefficient[Series[1/(1+ContinuedFractionK[Floor[(k^2+ 1)/2]*x*-1,1,{k,1,20}]),{x,0,20}],n],{n,1,20}]](* _Benedict W. J. Irwin_, Feb 10 2016 *)
%Y A005647 Cf. A000795, A000982.
%K A005647 nonn,easy,nice
%O A005647 0,3
%A A005647 _Simon Plouffe_, _N. J. A. Sloane_