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A012085 Even coefficients in expansion of e.g.f. cos(x)/sqrt(cos(2*x)). 3

%I #20 Oct 02 2023 13:48:30

%S 1,1,17,721,58337,7734241,1526099057,419784870961,153563504618177,

%T 72104198836466881,42270463533824671697,30262124466958766778001,

%U 25981973075048213029395617,26350476755161831091778460321

%N Even coefficients in expansion of e.g.f. cos(x)/sqrt(cos(2*x)).

%F E.g.f.: Sum_{k>=0} a(k)x^(2k)/(2k)! = cos(x)/sqrt(cos(2*x)) = sec(arcsin(tan(x))).

%F a(n) ~ 2*sqrt(2/Pi) * n^(2*n) * (8/Pi)^(2*n) / exp(2*n). - _Vaclav Kotesovec_, Oct 07 2013

%F a(n) = Sum_{j=0..n} Sum_{k=0..j} (2*n+1)!*(4*k-2*j+1)^(2*n)/(n!*(n-j)!*k!*(j-k)!*(2*j+1)*(-2)^j*(-4)^n). - _Tani Akinari_, Oct 02 2023

%e sec(arcsin(tan(x))) = 1 + 1/2!*x^2 + 17/4!*x^4 + 721/6!*x^6 + 58337/8!*x^8...

%t Table[n!*SeriesCoefficient[Cos[x]/Sqrt[Cos[2*x]],{x,0,n}],{n,0,30,2}] (* _Vaclav Kotesovec_, Oct 07 2013 *)

%o (PARI) {a(n)=local(A); if(n<0, 0, n*=2; A=x*O(x^n); n!*polcoeff( cos(x+A)/sqrt(cos(2*x+A)), n))} /* _Michael Somos_, Jul 18 2005 */

%o (PARI) {a(n)=sum(j=0,n,sum(k=0,j,(2*n+1)!*(4*k-2*j+1)^(2*n)/(n!*(n-j)!*k!*(j-k)!*(2*j+1)*(-2)^j*(-4)^n)))}; /* _Tani Akinari_, Oct 02 2023 */

%K nonn

%O 0,3

%A Patrick Demichel (patrick.demichel(AT)hp.com)

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Last modified June 14 04:49 EDT 2024. Contains 373393 sequences. (Running on oeis4.)