A273031 Expansion of e.g.f.: (sin(x) + sin(6*x)) / sin(7*x), even-indexed terms only.

1, 6, 330, 48726, 13534410, 6046913046, 3962771924490, 3580686141374166, 4266519857080266570, 6481738795978992136086, 12228451239686387772736650, 28048508112504152087554462806, 76867928701091608252297826870730, 248058932215537567368765344245378326, 931049990613171839116868739409352364810, 4021504762182514582910341826029900914866646

Offset

0,2

Links

Table of n, a(n) for n=0..15.

Formula

E.g.f.: cos(5*x/2) / cos(7*x/2).

E.g.f.: (cos(x) + cos(6*x)) / (1 + cos(7*x)).

E.g.f.: (exp(i*x) + exp(6*i*x)) / (1 + exp(7*i*x)), where i^2 = -1.

E.g.f.: exp(i*x)/(1 + exp(7*i*x)) + exp(-i*x)/(1 + exp(-7*i*x)), where i^2 = -1.

O.g.f.: 1/(1 - 1*6*x/(1 - 7^2*x/(1 - 8*13*x/(1 - 14^2*x/(1 - ... - (7*n+1)*(7*n+6)*x/(1 - (7*n+7)^2*x/(1 - ...))))))), a continued fraction.

a(n) ~ (2*n)! * 4*cos(5*Pi/14) * 7^(2*n) / Pi^(2*n+1). - Vaclav Kotesovec, May 14 2016

a(n) = (-49)^n*Euler(2*n, 1/7). - Peter Luschny, Nov 26 2020

Example

E.g.f.: A(x) = 1 + 6*x^2/2! + 330*x^4/4! + 48726*x^6/6! + 13534410*x^8/8! + 6046913046*x^10/10! + 3962771924490*x^12/12! + 3580686141374166*x^14/14! +...
such that A(x) = (sin(x) + sin(6*x)) / sin(7*x).
O.g.f.: F(x) = 1 + 6*x + 330*x^2 + 48726*x^3 + 13534410*x^4 + 6046913046*x^5 + 3962771924490*x^6 + 3580686141374166*x^7 +...
such that the o.g.f. can be expressed as the continued fraction:
F(x) = 1/(1 - 1*6*x/(1 - 7^2*x/(1 - 8*13*x/(1 - 14^2*x/(1 - 15*20*x/(1 - 21^2*x/(1 - 22*27*x/(1 - 28^2*x/(1 - 29*34*x/(1 - 35^2*x/(1 - 36*41*x/(1 - ...)))))))))))).

Maple

seq((-49)^n*euler(2*n, 1/7), n = 0..15); # Peter Luschny, Nov 26 2020

Mathematica

With[{nn=30}, Take[CoefficientList[Series[(Sin[x]+Sin[6x])/Sin[7x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Jul 08 2018 *)

Prog

(PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(1*X) + sin(6*X))/sin(7*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(1*X) + cos(6*X))/(1 + cos(7*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(1*I*X) + exp(6*I*X))/(1 + exp(7*I*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A272158, A272467, A273032, A273033, A156191.

Sequence in context: A003742 A324099 A135195 * A358485 A001509 A295925

Adjacent sequences: A273028 A273029 A273030 * A273032 A273033 A273034

Keyword

nonn

Author

Paul D. Hanna, May 13 2016

Status

approved