A272468 E.g.f.: ( (sin(2*x) + sin(3*x)) / sin(5*x) )^(1/6).

1, 1, 16, 861, 96151, 18222146, 5239250961, 2125867405481, 1156996954702696, 813362896424049741, 717389213154874345231, 775695142663748111834426, 1009031532010773852853587441, 1554520965241408817492939532161, 2799176143277347317623990785312576, 5825065298299069164298296125454811821, 13872866932424152546975929055708940259511, 37490505378893715802821349609594581921197906

Offset

0,3

Comments

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)).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

E.g.f.: ( (cos(2*x) + cos(3*x)) / (1 + cos(5*x)) )^(1/6).

E.g.f.: ( (exp(2*i*x) + exp(3*i*x)) / (1 + exp(5*i*x)) )^(1/6), where i^2 = -1.

a(n) = 1 (mod 5) for n>0.

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

Example

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! +...
RELATED SERIES.
The logarithm of the e.g.f. begins:
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! +...
such that the coefficients of x^(2*n)/(2*n)! consist entirely of odd integers.

Prog

(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)}
for(n=0, 20, print1(a(n), ", "))
(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)}
for(n=0, 20, print1(a(n), ", "))
(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)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A272467.

Sequence in context: A223113 A145243 A264125 * A214386 A367089 A185561

Adjacent sequences: A272465 A272466 A272467 * A272469 A272470 A272471

Keyword

nonn

Author

Paul D. Hanna, Apr 30 2016

Status

approved