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%I #16 Nov 26 2020 12:37:25
%S 1,6,330,48726,13534410,6046913046,3962771924490,3580686141374166,
%T 4266519857080266570,6481738795978992136086,
%U 12228451239686387772736650,28048508112504152087554462806,76867928701091608252297826870730,248058932215537567368765344245378326,931049990613171839116868739409352364810,4021504762182514582910341826029900914866646
%N Expansion of e.g.f.: (sin(x) + sin(6*x)) / sin(7*x), even-indexed terms only.
%F E.g.f.: cos(5*x/2) / cos(7*x/2).
%F E.g.f.: (cos(x) + cos(6*x)) / (1 + cos(7*x)).
%F E.g.f.: (exp(i*x) + exp(6*i*x)) / (1 + exp(7*i*x)), where i^2 = -1.
%F E.g.f.: exp(i*x)/(1 + exp(7*i*x)) + exp(-i*x)/(1 + exp(-7*i*x)), where i^2 = -1.
%F O.g.f.: 1/(1 - 1*6*x/(1 - 7^2*x/(1 - 8*13*x/(1 - 14^2*x/(1 - ... - (7*n+1)*(7*n+6)*x/(1 - (7*n+7)^2*x/(1 - ...))))))), a continued fraction.
%F a(n) ~ (2*n)! * 4*cos(5*Pi/14) * 7^(2*n) / Pi^(2*n+1). - _Vaclav Kotesovec_, May 14 2016
%F a(n) = (-49)^n*Euler(2*n, 1/7). - _Peter Luschny_, Nov 26 2020
%e E.g.f.: A(x) = 1 + 6*x^2/2! + 330*x^4/4! + 48726*x^6/6! + 13534410*x^8/8! + 6046913046*x^10/10! + 3962771924490*x^12/12! + 3580686141374166*x^14/14! +...
%e such that A(x) = (sin(x) + sin(6*x)) / sin(7*x).
%e O.g.f.: F(x) = 1 + 6*x + 330*x^2 + 48726*x^3 + 13534410*x^4 + 6046913046*x^5 + 3962771924490*x^6 + 3580686141374166*x^7 +...
%e such that the o.g.f. can be expressed as the continued fraction:
%e F(x) = 1/(1 - 1*6*x/(1 - 7^2*x/(1 - 8*13*x/(1 - 14^2*x/(1 - 15*20*x/(1 - 21^2*x/(1 - 22*27*x/(1 - 28^2*x/(1 - 29*34*x/(1 - 35^2*x/(1 - 36*41*x/(1 - ...)))))))))))).
%p seq((-49)^n*euler(2*n, 1/7), n = 0..15); # _Peter Luschny_, Nov 26 2020
%t With[{nn=30},Take[CoefficientList[Series[(Sin[x]+Sin[6x])/Sin[7x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Jul 08 2018 *)
%o (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(1*X) + sin(6*X))/sin(7*X), 2*n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (cos(1*X) + cos(6*X))/(1 + cos(7*X)), 2*n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (exp(1*I*X) + exp(6*I*X))/(1 + exp(7*I*X)), 2*n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A272158, A272467, A273032, A273033, A156191.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 13 2016
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