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A012085 Even coefficients in expansion of e.g.f. cos(x)/sqrt(cos(2*x)). 3
1, 1, 17, 721, 58337, 7734241, 1526099057, 419784870961, 153563504618177, 72104198836466881, 42270463533824671697, 30262124466958766778001, 25981973075048213029395617, 26350476755161831091778460321 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
E.g.f.: Sum_{k>=0} a(k)x^(2k)/(2k)! = cos(x)/sqrt(cos(2*x)) = sec(arcsin(tan(x))).
a(n) ~ 2*sqrt(2/Pi) * n^(2*n) * (8/Pi)^(2*n) / exp(2*n). - Vaclav Kotesovec, Oct 07 2013
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
EXAMPLE
sec(arcsin(tan(x))) = 1 + 1/2!*x^2 + 17/4!*x^4 + 721/6!*x^6 + 58337/8!*x^8...
MATHEMATICA
Table[n!*SeriesCoefficient[Cos[x]/Sqrt[Cos[2*x]], {x, 0, n}], {n, 0, 30, 2}] (* Vaclav Kotesovec, Oct 07 2013 *)
PROG
(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 */
(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 */
CROSSREFS
Sequence in context: A012029 A012193 A128274 * A298306 A308696 A308594
KEYWORD
nonn
AUTHOR
Patrick Demichel (patrick.demichel(AT)hp.com)
STATUS
approved

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Last modified May 22 21:38 EDT 2024. Contains 372758 sequences. (Running on oeis4.)