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

1, 6, 186, 14166, 2009946, 458225526, 153212718906, 70632832168086, 42939614599671066, 33282798350926545846, 32036398991671262130426, 37490905748197466281582806, 52420996658289450763331680986, 86309558223400912513674314622966, 165280246718130394753827229389826746, 364233987506387128128991081880073730326, 915234544675507984101674168382043517591706

Offset

0,2

Links

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

P. Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)

Formula

E.g.f.: cos(x/2) / cos(5*x/2).

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

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

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.

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.

a(n) = 6 (mod 10) for n>0.

a(n) ~ (2*n)! * sqrt(2*(5 + sqrt(5))) * 5^(2*n) / Pi^(2*n+1). - Vaclav Kotesovec, Apr 30 2016

From Peter Bala, May 13 2017: (Start)

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 - ....

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)

Example

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! +...
such that A(x) = (sin(2*x) + sin(3*x)) / sin(5*x).
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 +...
such that the o.g.f. can be expressed as the continued fraction:
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 - ...)))))))))))).

Mathematica

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 *)

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

Crossrefs

Cf. A272158, A272468, A156185.

Sequence in context: A175237 A222335 A037298 * A015004 A324094 A129046

Adjacent sequences: A272464 A272465 A272466 * A272468 A272469 A272470

Keyword

nonn

Author

Paul D. Hanna, Apr 30 2016

Status

approved