A263246 Expansion of e.g.f.: sin(r*x) / sqrt(1 + cos(r*x)^2) where r = sqrt(2), odd powers only.

1, 1, -11, -491, -11159, 460681, 103577629, 8160790429, -624333860399, -386787409545839, -68810049201689531, 6999828208693648549, 9872674440874152431161, 3255253386897615662908441, -346248578699462435167833491, -1072454627614122049417452882131, -584579592415141205182370782224479, 47874474639430619859527348515679521

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..234

Formula

a(n) = - A101922(n). - Michel Marcus, Sep 11 2022

Example

E.g.f.: S(x) = x + x^3/3! - 11*x^5/5! - 491*x^7/7! - 11159*x^9/9! + 460681*x^11/11! + 103577629*x^13/13! + 8160790429*x^15/15! +...
Related expansions.
S(x)^2 = 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 - S(x)^2) = 1 - x^2/2! - 7*x^4/4! - 49*x^6/6! + 1457*x^8/8! + 148799*x^10/10! + 6409193*x^12/12! +...+ A263247(n)*x^(2*n)/(2*n)! +...
sqrt(1 + S(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(2*n)*x^(2*n)/(2*n)! +...

Mathematica

r:= Sqrt[2]; With[{nmax = 500}, CoefficientList[Series[Sin[r*x]/Sqrt[1 + Cos[r*x]^2], {x, 0, nmax}], x]*Range[0, nmax - 1]!][[2 ;; -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+1)!*polcoeff(S, 2*n+1)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A263247, A263248, A263249.

Cf. A101922.

Sequence in context: A041931 A322277 A101922 * A263377 A142809 A139198

Adjacent sequences: A263243 A263244 A263245 * A263247 A263248 A263249

Keyword

sign

Author

Paul D. Hanna, Oct 13 2015

Status

approved