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

%I A263247 #9 Jul 27 2018 04:33:47
%S A263247 1,-1,-7,-49,1457,148799,6409193,-436948849,-155175606943,
%T A263247 -18245982604801,1864031151256793,1627915037217907151,
%U A263247 390178889220670506257,-46784571591411151243201,-89306450512551172914577207,-37461031331532428265812712049,4204976347690709918899169381057,17814701962096793952255775890393599
%N A263247 Expansion of e.g.f.: r*cos(r*x) / sqrt(1 + cos(r*x)^2)  where r = sqrt(2), even terms only.
%H A263247 G. C. Greubel, <a href="/A263247/b263247.txt">Table of n, a(n) for n = 0..234</a>
%e A263247 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! +...
%e A263247 Related expansions.
%e A263247 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)! +...
%e A263247 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)! +...
%e A263247 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)! +...
%t A263247 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 *)
%o A263247 (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)}
%o A263247 for(n=0,20,print1(a(n),", "))
%Y A263247 Cf. A263246, A263248, A263249.
%K A263247 sign
%O A263247 0,3
%A A263247 _Paul D. Hanna_, Oct 13 2015