A272158 Expansion of e.g.f.: (sin(x) + sin(4*x)) / sin(5*x), even-indexed terms only.

1, 4, 116, 8764, 1242356, 283202524, 94690800596, 43653497804284, 26538141745926836, 20569900661155862044, 19799583458238177373076, 23170654021955185224223804, 32397957659053038859810291316, 53342240536065395589518876137564, 102148810140776173440241789042633556, 225108984136852617968906778958292851324, 565646056287498262815832721506444163551796

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..208

Formula

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

E.g.f.: (cos(x) + cos(4*x)) / (1 + cos(5*x)).

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

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

O.g.f.: 1/(1 - 1*4*x/(1 - 5^2*x/(1 - 6*9*x/(1 - 10^2*x/(1 - ... - (5*n+1)*(5*n+4)*x/(1 - (5*n+5)^2*x/(1 - ...))))))), a continued fraction.

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

a(n) = (-25)^n*Euler(2*n, 1/5). - Peter Luschny, Nov 26 2020

Example

E.g.f.: A(x) = 1 + 4*x^2/2! + 116*x^4/4! + 8764*x^6/6! + 1242356*x^8/8! + 283202524*x^10/10! + 94690800596*x^12/12! +...
such that A(x) = (sin(x) + sin(4*x)) / sin(5*x).
O.g.f.: F(x) = 1 + 4*x + 116*x^2 + 8764*x^3 + 1242356*x^4 + 283202524*x^5 + 94690800596*x^6 + 43653497804284*x^7 + 26538141745926836*x^8 +...
such that the o.g.f. can be expressed as the continued fraction:
F(x) = 1/(1 - 1*4*x/(1 - 5^2*x/(1 - 6*9*x/(1 - 10^2*x/(1 - 11*14*x/(1 - 15^2*x/(1 - 16*19*x/(1 - 20^2*x/(1 - 21*24*x/(1 - 25^2*x/(1 - 26*29*x/(1 - ...)))))))))))).

Maple

seq((-25)^n*euler(2*n, 1/5), n = 0..16); # Peter Luschny, Nov 26 2020

Mathematica

Table[(CoefficientList[Series[(Sin[x] + Sin[4*x]) / Sin[5*x], {x, 0, 40}], x]*Range[0, 40]!)[[2*n + 1]], {n, 0, 20}] (* Vaclav Kotesovec, Apr 30 2016 *)
With[{nmax = 60}, CoefficientList[Series[Cos[3*x/2]/Cos[5*x/2], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]] (* G. C. Greubel, Oct 11 2018 *)

Prog

(PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(X) + sin(4*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(X) + cos(4*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(I*X) + exp(4*I*X))/(1 + exp(5*I*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A272467.

Sequence in context: A340277 A206689 A198080 * A194535 A030255 A146508

Adjacent sequences: A272155 A272156 A272157 * A272159 A272160 A272161

Keyword

nonn

Author

Paul D. Hanna, Apr 30 2016

Status

approved