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A263248
E.g.f.: r / sqrt(1 + cos(r*x)^2) where r = sqrt(2), even powers only.
4
1, 1, 1, -71, -2591, -23759, 7872481, 1032165289, 34225547329, -19224419375519, -5800472581083839, -474084524873544551, 353987939065905654049, 201460031539970745643921, 32857189444574660214635041, -29238884957420392451016521591, -28126153570109708198511424386431, -8022417111018145463775521643973439, 7957314358326789159275513256441813121
OFFSET
0,4
LINKS
EXAMPLE
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! + ...
Related expansions.
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)! + ...
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)! + ...
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)! + ...
MATHEMATICA
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 *)
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 *)
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(D, 2*n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 13 2015
STATUS
approved