|
%I #15 Apr 30 2016 11:01:09
%S 1,1,16,861,96151,18222146,5239250961,2125867405481,1156996954702696,
%T 813362896424049741,717389213154874345231,775695142663748111834426,
%U 1009031532010773852853587441,1554520965241408817492939532161,2799176143277347317623990785312576,5825065298299069164298296125454811821,13872866932424152546975929055708940259511,37490505378893715802821349609594581921197906
%N E.g.f.: ( (sin(2*x) + sin(3*x)) / sin(5*x) )^(1/6).
%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)).
%H Paul D. Hanna, <a href="/A272468/b272468.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: ( (cos(2*x) + cos(3*x)) / (1 + cos(5*x)) )^(1/6).
%F E.g.f.: ( (exp(2*i*x) + exp(3*i*x)) / (1 + exp(5*i*x)) )^(1/6), where i^2 = -1.
%F a(n) = 1 (mod 5) for n>0.
%F a(n) ~ (2*n)! * (2*(5 + sqrt(5)))^(1/12) * 5^(2*n) / (Gamma(1/6) * Pi^(2*n + 1/6) * n^(5/6)). - _Vaclav Kotesovec_, Apr 30 2016
%e G.f.: A(x) = 1 + x^2/2! + 16*x^4/4! + 861*x^6/6! + 96151*x^8/8! + 18222146*x^10/10! + 5239250961*x^12/12! + 2125867405481*x^14/14! +...
%e RELATED SERIES.
%e The logarithm of the e.g.f. begins:
%e log(A(x)) = x^2/2! + 13*x^4/4! + 651*x^6/6! + 69173*x^8/8! + 12613931*x^10/10! + 3514607733*x^12/12! + 1388804117611*x^14/14! + 738755067184693*x^16/16! + 508990446726347691*x^18/18! + 440936448176697240053*x^20/20! +...
%e such that the coefficients of x^(2*n)/(2*n)! consist entirely of odd integers.
%o (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))^(1/6), 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(2*X) + cos(3*X))/(1 + cos(5*X)) )^(1/6), 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(2*I*X) + exp(3*I*X))/(1 + exp(5*I*X)))^(1/6), 2*n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A272467.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 30 2016
| 