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A272467 E.g.f.: (sin(2*x) + sin(3*x)) / sin(5*x).

%I A272467 #41 Jun 12 2022 16:58:16
%S A272467 1,6,186,14166,2009946,458225526,153212718906,70632832168086,
%T A272467 42939614599671066,33282798350926545846,32036398991671262130426,
%U A272467 37490905748197466281582806,52420996658289450763331680986,86309558223400912513674314622966,165280246718130394753827229389826746,364233987506387128128991081880073730326,915234544675507984101674168382043517591706
%N A272467 E.g.f.: (sin(2*x) + sin(3*x)) / sin(5*x).
%H A272467 Paul D. Hanna, <a href="/A272467/b272467.txt">Table of n, a(n) for n = 0..200</a>
%H A272467 P. Bala, <a href="/A002439/a002439.pdf">Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)</a>
%F A272467 E.g.f.: cos(x/2) / cos(5*x/2).
%F A272467 E.g.f.: (cos(2*x) + cos(3*x)) / (1 + cos(5*x)).
%F A272467 E.g.f.: (exp(2*i*x) + exp(3*i*x)) / (1 + exp(5*i*x)), where i^2 = -1.
%F A272467 E.g.f.: exp(2*i*x)/(1 + exp(5*i*x)) + exp(-2*i*x)/(1 + exp(-5*i*x)), where i^2 = -1.
%F A272467 O.g.f.: 1/(1 - 2*3*x/(1 - 5^2*x/(1 - 7*8*x/(1 - 10^2*x/(1 - ... - (5*n+2)*(5*n+3)*x/(1 - (5*n+5)^2*x/(1 - ...))))))), a continued fraction.
%F A272467 a(n) = 6 (mod 10) for n>0.
%F A272467 a(n) ~ (2*n)! * sqrt(2*(5 + sqrt(5))) * 5^(2*n) / Pi^(2*n+1). - _Vaclav Kotesovec_, Apr 30 2016
%F A272467 From _Peter Bala_, May 13 2017: (Start)
%F A272467 G.f.: 1/(1 + 4*x - 10*x/(1 - 15*x/(1 + 4*x - 70*x/(1 - 80*x/(1 + 4*x - ... - 5*n*(5*n-3)*x/(1 - 5*n*(5*n-2)*x/(1 + 4*x - ....
%F A272467 G.f.: 1/(1 + 9*x - 15*x/(1 - 10*x/(1 + 9*x - 80*x/(1 - 70*x/(1 + 9*x - ... - 5*n*(5*n-2)*x/(1 - 5*n*(5*n-3)*x/(1 + 9*x - .... (End)
%e A272467 E.g.f.: A(x) = 1 + 6*x^2/2! + 186*x^4/! + 14166*x^6/6! + 2009946*x^8/8! + 458225526*x^10/10! + 153212718906*x^12/12! +...
%e A272467 such that A(x) = (sin(2*x) + sin(3*x)) / sin(5*x).
%e A272467 O.g.f.: F(x) = 1 + 6*x + 186*x^2 + 14166*x^3 + 2009946*x^4 + 458225526*x^5 + 153212718906*x^6 + 70632832168086*x^7 + 42939614599671066*x^8 +...
%e A272467 such that the o.g.f. can be expressed as the continued fraction:
%e A272467 F(x) = 1/(1 - 2*3*x/(1 - 5^2*x/(1 - 7*8*x/(1 - 10^2*x/(1 - 12*13*x/(1 - 15^2*x/(1 - 17*18*x/(1 - 20^2*x/(1 - 22*23*x/(1 - 25^2*x/(1 - 27*28*x/(1 - ...)))))))))))).
%t A272467 With[{nn=40},Take[CoefficientList[Series[(Sin[2x]+Sin[3x])/Sin[5x],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Jun 12 2022 *)
%o A272467 (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(2*X) + sin(3*X))/sin(5*X), 2*n)}
%o A272467 for(n=0, 20, print1(a(n), ", "))
%o A272467 (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (cos(2*X) + cos(3*X))/(1 + cos(5*X)), 2*n)}
%o A272467 for(n=0, 20, print1(a(n), ", "))
%o A272467 (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (exp(2*I*X) + exp(3*I*X))/(1 + exp(5*I*X)), 2*n)}
%o A272467 for(n=0, 20, print1(a(n), ", "))
%Y A272467 Cf. A272158, A272468, A156185.
%K A272467 nonn
%O A272467 0,2
%A A272467 _Paul D. Hanna_, Apr 30 2016