A263247 Expansion of e.g.f.: r*cos(r*x) / sqrt(1 + cos(r*x)^2) where r = sqrt(2), even terms only.

1, -1, -7, -49, 1457, 148799, 6409193, -436948849, -155175606943, -18245982604801, 1864031151256793, 1627915037217907151, 390178889220670506257, -46784571591411151243201, -89306450512551172914577207, -37461031331532428265812712049, 4204976347690709918899169381057, 17814701962096793952255775890393599

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..234

Example

E.g.f.: C(x) = 1 - x^2/2! - 7*x^4/4! - 49*x^6/6! + 1457*x^8/8! + 148799*x^10/10! + 6409193*x^12/12! - 436948849*x^14/14! +...
Related expansions.
C(x)^2 = 1 - 2*x^2/2! - 8*x^4/4! + 112*x^6/6! + 9088*x^8/8! + 310528*x^10/10! - 14701568*x^12/12! +...+ A263249(n)*x^(2*n)/(2*n)! +...
sqrt(1 - C(x)^2) = x + x^3/3! - 11*x^5/5! - 491*x^7/7! - 11159*x^9/9! + 460681*x^11/11! +...+ A263246(n)*x^(2*n+1)/(2*n+1)! +...
sqrt(2 - C(x)^2) = 1 + x^2/2! + x^4/4! - 71*x^6/6! - 2591*x^8/8! - 23759*x^10/10! + 7872481*x^12/12! +...+ A263248(n)*x^(2*n)/(2*n)! +...

Mathematica

r:= Sqrt[2]; With[{nmax = 60}, CoefficientList[Series[r*Cos[r*x]/Sqrt[1 + Cos[r*x]^2], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; -1 ;; 2]] (* G. C. Greubel, Jul 27 2018 *)

Prog

(PARI) {a(n) = local(S=x, C=1, D=1, ox=O(x^(2*n+2))); for(i=1, 2*n+1, S = intformal(C*D^2 +ox); C = 1 - intformal(S*D^2); D = 1 + intformal(S*C*D); ); (2*n)!*polcoeff(C, 2*n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A263246, A263248, A263249.

Sequence in context: A080895 A047899 A229041 * A259990 A061585 A202782

Adjacent sequences: A263244 A263245 A263246 * A263248 A263249 A263250

Keyword

sign

Author

Paul D. Hanna, Oct 13 2015

Status

approved