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A263248 E.g.f.: r / sqrt(1 + cos(r*x)^2) where r = sqrt(2), even powers only.

%I A263248 #14 Jul 27 2018 09:36:49
%S A263248 1,1,1,-71,-2591,-23759,7872481,1032165289,34225547329,
%T A263248 -19224419375519,-5800472581083839,-474084524873544551,
%U A263248 353987939065905654049,201460031539970745643921,32857189444574660214635041,-29238884957420392451016521591,-28126153570109708198511424386431,-8022417111018145463775521643973439,7957314358326789159275513256441813121
%N A263248 E.g.f.: r / sqrt(1 + cos(r*x)^2)  where r = sqrt(2), even powers only.
%H A263248 G. C. Greubel, <a href="/A263248/b263248.txt">Table of n, a(n) for n = 0..234</a>
%e A263248 E.g.f.: D(x) = 1 + x^2/2! + x^4/4! - 71*x^6/6! - 2591*x^8/8! - 23759*x^10/10! + 7872481*x^12/12! + 1032165289*x^14/14! + ...
%e A263248 Related expansions.
%e A263248 D(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)! + ...
%e A263248 sqrt(D(x)^2 - 1) = 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)! + ...
%e A263248 sqrt(2 - D(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)! + ...
%t A263248 Table[SeriesCoefficient[Series[EllipticF[x,1/2],{x,0,41}],2n+1](2n+1)!2^n, {n, 0, 20}] (* _Benedict W. J. Irwin_, Apr 06 2017 *)
%t A263248 r:= Sqrt[2]; With[{nmax = 60}, CoefficientList[Series[r/Sqrt[1 + Cos[r*x]^2], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; -1 ;; 2]] (* _G. C. Greubel_, Jul 27 2018 *)
%o A263248 (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(D, 2*n)}
%o A263248 for(n=0,20,print1(a(n),", "))
%Y A263248 Cf. A263246, A263247, A263249.
%K A263248 sign
%O A263248 0,4
%A A263248 _Paul D. Hanna_, Oct 13 2015