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A272158 Expansion of e.g.f.: (sin(x) + sin(4*x)) / sin(5*x), even-indexed terms only.

%I A272158 #37 Nov 26 2020 15:56:44
%S A272158 1,4,116,8764,1242356,283202524,94690800596,43653497804284,
%T A272158 26538141745926836,20569900661155862044,19799583458238177373076,
%U A272158 23170654021955185224223804,32397957659053038859810291316,53342240536065395589518876137564,102148810140776173440241789042633556,225108984136852617968906778958292851324,565646056287498262815832721506444163551796
%N A272158 Expansion of e.g.f.: (sin(x) + sin(4*x)) / sin(5*x), even-indexed terms only.
%H A272158 G. C. Greubel, <a href="/A272158/b272158.txt">Table of n, a(n) for n = 0..208</a>
%F A272158 E.g.f.: cos(3*x/2) / cos(5*x/2).
%F A272158 E.g.f.: (cos(x) + cos(4*x)) / (1 + cos(5*x)).
%F A272158 E.g.f.: (exp(i*x) + exp(4*i*x)) / (1 + exp(5*i*x)), where i^2 = -1.
%F A272158 E.g.f.: exp(i*x)/(1 + exp(5*i*x)) + exp(-i*x)/(1 + exp(-5*i*x)), where i^2 = -1.
%F A272158 O.g.f.: 1/(1 - 1*4*x/(1 - 5^2*x/(1 - 6*9*x/(1 - 10^2*x/(1 - ... - (5*n+1)*(5*n+4)*x/(1 - (5*n+5)^2*x/(1 - ...))))))), a continued fraction.
%F A272158 a(n) ~ (2*n)! * sqrt(2*(5 - sqrt(5))) * 5^(2*n) / Pi^(2*n+1). - _Vaclav Kotesovec_, Apr 30 2016
%F A272158 a(n) = (-25)^n*Euler(2*n, 1/5). - _Peter Luschny_, Nov 26 2020
%e A272158 E.g.f.: A(x) = 1 + 4*x^2/2! + 116*x^4/4! + 8764*x^6/6! + 1242356*x^8/8! + 283202524*x^10/10! + 94690800596*x^12/12! +...
%e A272158 such that A(x) = (sin(x) + sin(4*x)) / sin(5*x).
%e A272158 O.g.f.: F(x) = 1 + 4*x + 116*x^2 + 8764*x^3 + 1242356*x^4 + 283202524*x^5 + 94690800596*x^6 + 43653497804284*x^7 + 26538141745926836*x^8 +...
%e A272158 such that the o.g.f. can be expressed as the continued fraction:
%e A272158 F(x) = 1/(1 - 1*4*x/(1 - 5^2*x/(1 - 6*9*x/(1 - 10^2*x/(1 - 11*14*x/(1 - 15^2*x/(1 - 16*19*x/(1 - 20^2*x/(1 - 21*24*x/(1 - 25^2*x/(1 - 26*29*x/(1 - ...)))))))))))).
%p A272158 seq((-25)^n*euler(2*n, 1/5), n = 0..16); # _Peter Luschny_, Nov 26 2020
%t A272158 Table[(CoefficientList[Series[(Sin[x] + Sin[4*x]) / Sin[5*x], {x, 0, 40}], x]*Range[0, 40]!)[[2*n + 1]], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 30 2016 *)
%t A272158 With[{nmax = 60}, CoefficientList[Series[Cos[3*x/2]/Cos[5*x/2], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]] (* _G. C. Greubel_, Oct 11 2018 *)
%o A272158 (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(X) + sin(4*X))/sin(5*X), 2*n)}
%o A272158 for(n=0, 20, print1(a(n), ", "))
%o A272158 (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (cos(X) + cos(4*X))/(1 + cos(5*X)), 2*n)}
%o A272158 for(n=0, 20, print1(a(n), ", "))
%o A272158 (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (exp(I*X) + exp(4*I*X))/(1 + exp(5*I*X)), 2*n)}
%o A272158 for(n=0, 20, print1(a(n), ", "))
%Y A272158 Cf. A272467.
%K A272158 nonn
%O A272158 0,2
%A A272158 _Paul D. Hanna_, Apr 30 2016