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Numerators of sequence having sqrt(cos(x)) as e.g.f. (even-indexed coefficients only).
1

%I #53 Apr 15 2022 13:02:16

%S 1,-1,-1,-19,-559,-29161,-2368081,-276580459,-43947282079,

%T -9118829535121,-2394495729300961,-776228170682260099,

%U -304471093666800990799,-142128398853646068197881,-77865168574139358455774641,-49474260304294496117945326939

%N Numerators of sequence having sqrt(cos(x)) as e.g.f. (even-indexed coefficients only).

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.

%D H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 206.

%H Vincenzo Librandi, <a href="/A008990/b008990.txt">Table of n, a(n) for n = 0..100</a>

%F E.g.f.: sqrt(cos(sqrt(2)*x)) = 1 - x^2/2! - x^4/4! - 19x^6/6! -... - _Ralf Stephan_, Mar 03 2005

%F a(n) = sum(sum(binomial(k,j)*2^(n+2-2*k-j)*sum(binomial(j,i)*(j-2*i)^(2*n), i=0..floor((j-1)/2))*(-1)^(n+j+1), j=1..k)*C(k-1), k=1..2*n), n>0, C(n) - Catalan numbers (A000108). - _Vladimir Kruchinin_, Sep 10 2010

%F G.f.: 2/G(0) where G(k) = 2 - 4*x*(k+1)*(2*k-1)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 12 2013

%F G.f.: Q(0), where Q(k) = 1 - x*(k+1)*(2*k-1)/( x*(k+1)*(2*k-1) + 2/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 21 2013

%F a(n) = (1/(4*n))*Sum_{k=1..n} C(2*n,2*k)*(-1)^(k)*2^(3*k)*(2^(2*k)-1)*B(2*k)*a(n-k), a(0)=1, where B(n) is Bernoulli numbers. - _Vladimir Kruchinin_, Jun 23 2015.

%F The odd terms of EllipticE(x,2) act as a g.f. for 2^n*a(n)/(2*n+1)!. - _Benedict W. J. Irwin_, Jun 06 2016

%F a(n) ~ -2^(5*n - 1) * n^(2*n - 1) / (Pi^(2*n - 1/2) * exp(2*n)). - _Vaclav Kotesovec_, Jun 11 2016

%F O.g.f. as a continued fraction of Stieltjes type: 1/(1 + x/(1 - 2*x/(1 - 9*x/(1 - 20*x/(1 - ... - (2*n^2 + n - 1)*x/(1 - ... )))))) = 1 - x - x^2 - 19*x^3 - 559*x^4 - .... Follows from Wall, equation 53.11, p. 206 with k = - 1/2. - _Peter Bala_, Apr 11 2022

%t n = 32; Partition[ CoefficientList[ Series[ Sqrt[Cos[Sqrt[2]*x]], {x, 0, n}], x]*Range[0, n]!, 2][[All, 1]] (* _Jean-François Alcover_, Aug 30 2011 *)

%t Table[SeriesCoefficient[Series[EllipticE[x, 2], {x, 0, 41}], 2 n + 1] (2 n + 1)!/2^n, {n, 0, 20}] (* _Benedict W. J. Irwin_, Jun 06 2016 *)

%o (Maxima) C(n):=1/(n+1)*binomial(2*n,n); a(n):=sum(sum(binomial(k,j) *2^(n+2-2*k-j)*sum(binomial(j,i)*(j-2*i)^(2*n),i,0,floor((j-1)/2))*(-1)^(n+j+1),j,1,k)*C(k-1),k,1,2*n); /* _Vladimir Kruchinin_, Sep 10 2010 */

%o (Maxima) a(n):=if n=0 then 1 else 1/(4*n)*sum(binomial(2*n,2*k)*(-1)^(k)*2^(3*k)*(2^(2*k)-1)*bern(2*k)*a(n-k),k,1,n); /* _Vladimir Kruchinin_, Jun 23 2015 */

%Y Cf. A027641.

%Y Denominators are in A000079.

%K sign,frac,easy,nice

%O 0,4

%A _N. J. A. Sloane_