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%I #17 Sep 23 2019 11:37:06
%S 1,12,732,109332,30406812,13587056052,8904250650492,8045727017033172,
%T 9586782871360007772,14564334832981893064692,
%U 27477080512619965247054652,63024425641459625896776174612,172720667970739808701108304367132,557383361208023769780400587942586932,2092050338949043346342979863638489321212,9036239176876728629700436615577988154925652
%N E.g.f.: (sin(3*x) + sin(4*x)) / sin(7*x).
%H 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 E.g.f.: cos(x/2) / cos(7*x/2).
%F E.g.f.: (cos(3*x) + cos(4*x)) / (1 + cos(7*x)).
%F E.g.f.: (exp(3*i*x) + exp(4*i*x)) / (1 + exp(7*i*x)), where i^2 = -1.
%F E.g.f.: exp(3*i*x)/(1 + exp(7*i*x)) + exp(-3*i*x)/(1 + exp(-7*i*x)), where i^2 = -1.
%F O.g.f.: 1/(1 - 3*4*x/(1 - 7^2*x/(1 - 10*11*x/(1 - 14^2*x/(1 - ... - (7*n+3)*(7*n+4)*x/(1 - (7*n+7)^2*x/(1 - ...))))))), a continued fraction.
%F a(n) ~ (2*n)! * 4*cos(Pi/14) * 7^(2*n) / Pi^(2*n+1). - _Vaclav Kotesovec_, May 14 2016
%F From _Peter Bala_, May 13 2017: (Start)
%F G.f.: 1/(1 + 9*x - 21*x/(1 - 28*x/(1 + 9*x - 140*x/(1 - 154*x/(1 + 9*x - ... - 7*n*(7*n-4)*x/(1 - 7*n*(7*n-3)*x/(1 + 9*x - ...
%F G.f.: 1/(1 + 16*x - 28*x/(1 - 21*x/(1 + 16*x - 154*x/(1 - 140*x/(1 + 16*x - ... - 7*n*(7*n-3)*x/(1 - 7*n*(7*n-4)*x/(1 + 16*x - .... (End)
%e E.g.f.: A(x) = 1 + 12*x^2/2! + 732*x^4/4! + 109332*x^6/6! + 30406812*x^8/8! + 13587056052*x^10/10! + 8904250650492*x^12/12! +...
%e such that A(x) = (sin(3*x) + sin(4*x)) / sin(7*x).
%e O.g.f.: F(x) = 1 + 12*x + 732*x^2 + 109332*x^3 + 30406812*x^4 + 13587056052*x^5 + 8904250650492*x^6 + 8045727017033172*x^7 +...
%e such that the o.g.f. can be expressed as the continued fraction:
%e F(x) = 1/(1 - 3*4*x/(1 - 7^2*x/(1 - 10*11*x/(1 - 14^2*x/(1 - 17*18*x/(1 - 21^2*x/(1 - 24*25*x/(1 - 28^2*x/(1 - 31*32*x/(1 - 35^2*x/(1 - 38*39*x/(1 - ...)))))))))))).
%t With[{nn=40},Take[CoefficientList[Series[(Sin[3x]+Sin[4x])/Sin[7x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Sep 23 2019 *)
%o (PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(3*X) + sin(4*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(3*X) + cos(4*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(3*I*X) + exp(4*I*X))/(1 + exp(7*I*X)), 2*n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A272158, A272467, A273031, A273032, A156196.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 13 2016
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