A263246 - OEIS

%I #17 Sep 11 2022 09:28:30

%S 1,1,-11,-491,-11159,460681,103577629,8160790429,-624333860399,

%T -386787409545839,-68810049201689531,6999828208693648549,

%U 9872674440874152431161,3255253386897615662908441,-346248578699462435167833491,-1072454627614122049417452882131,-584579592415141205182370782224479,47874474639430619859527348515679521

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

%H G. C. Greubel, <a href="/A263246/b263246.txt">Table of n, a(n) for n = 1..234</a>

%F a(n) = - A101922(n). - _Michel Marcus_, Sep 11 2022

%e 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! +...

%e Related expansions.

%e 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)! +...

%e 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)! +...

%e 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)! +...

%t 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 *)

%o (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)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A263247, A263248, A263249.

%Y Cf. A101922.

%K sign

%O 1,3

%A _Paul D. Hanna_, Oct 13 2015