A272462 - OEIS

%I #18 Apr 30 2016 11:15:37

%S 1,144,110736,210615984,746983671696,4257350521714224,

%T 35587232523796298256,410153047417991816658864,

%U 6233578942096421437956406416,120792275605361900683656723353904,2906718881167945096170286249866172176,85040397046127662408816208740192211143344,2972655821721881151201962319719058551015131536,122359658857407586222518613315327725437983909211184

%N E.g.f.: (sin(3^2*x) + sin(4^2*x)) / sin(5^2*x).

%C Conjecture: Given positive integers a and b, then the coefficient of x^(2*n)/(2*n)! is integral for n>=0 in the power series expansion of ( (sin(a*x) + sin(b*x)) / sin((a+b)*x) )^(1/(a*b)).

%F E.g.f.: (cos(3^2*x) + cos(4^2*x))/(1 + cos(5^2*x)).

%F E.g.f.: (exp(3^2*i*x) + exp(4^2*i*x))/(1 + exp(5^2*i*x)), where i^2 = -1.

%F O.g.f.: 1/(1 - 9*16*x/(1 - 25^2*x/(1 - 34*41*x/(1 - 50^2*x/(1 - 59*66*x/(1 - 75^2*x/(1 - 84*91*x/(1 - 100^2*x/(1 - 109*116*x/(1 - 125^2*x/(1 - 134*141*x/(1 - ...)))))))))))), a continued fraction.

%F a(n) ~ (2*n)! * 4*cos(7*Pi/50) * 5^(4*n) / Pi^(2*n+1). - _Vaclav Kotesovec_, Apr 30 2016

%e E.g.f.: A(x) = 1 + 144*x^2/2! + 110736*x^4/4! + 210615984*x^6/6! + 746983671696*x^8/8! + 4257350521714224*x^10/10! + 35587232523796298256*x^12/12! +...

%e where A(x) = (sin(9*x) + sin(16*x)) / sin(25*x).

%e RELATED SERIES.

%e The 144-th root of the e.g.f. results in an e.g.f. with integer coefficients of x^(2*n)/(2*n)! for n>=0, as illustrated by:

%e A(x)^(1/144) = 1 + x^2/2! + 340*x^4/4! + 428721*x^6/6! + 1141730995*x^8/8! + 5213697635366*x^10/10! + 36365790055623225*x^12/12! + 359653600595667938561*x^14/14! + 4787444684141055953638780*x^16/16! + + 82531879011717976623220644681*x^18/18! + 1788779716747879258524044670650035*x^20/20! +...

%t Table[(CoefficientList[Series[(Sin[9*x] + Sin[16*x]) / Sin[25*x], {x, 0, 40}], x] * Range[0, 40]!)[[2*n + 1]], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 30 2016 *)

%o (PARI) {a(n) = my(A=1,X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(3^2*X) + sin(4^2*X))/sin(5^2*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(3^2*I*X) + exp(4^2*I*X))/(1 + exp(5^2*I*X)), 2*n)}

%o for(n=0,20, print1(a(n),", "))

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 29 2016