A273033 E.g.f.: (sin(3*x) + sin(4*x)) / sin(7*x).

1, 12, 732, 109332, 30406812, 13587056052, 8904250650492, 8045727017033172, 9586782871360007772, 14564334832981893064692, 27477080512619965247054652, 63024425641459625896776174612, 172720667970739808701108304367132, 557383361208023769780400587942586932, 2092050338949043346342979863638489321212, 9036239176876728629700436615577988154925652

Offset

0,2

Links

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

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(7*x/2).

E.g.f.: (cos(3*x) + cos(4*x)) / (1 + cos(7*x)).

E.g.f.: (exp(3*i*x) + exp(4*i*x)) / (1 + exp(7*i*x)), where i^2 = -1.

E.g.f.: exp(3*i*x)/(1 + exp(7*i*x)) + exp(-3*i*x)/(1 + exp(-7*i*x)), where i^2 = -1.

O.g.f.: 1/(1 - 3*4*x/(1 - 7^2*x/(1 - 10*11*x/(1 - 14^2*x/(1 - ... - (7*n+3)*(7*n+4)*x/(1 - (7*n+7)^2*x/(1 - ...))))))), a continued fraction.

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

From Peter Bala, May 13 2017: (Start)

G.f.: 1/(1 + 9*x - 21*x/(1 - 28*x/(1 + 9*x - 140*x/(1 - 154*x/(1 + 9*x - ... - 7*n*(7*n-4)*x/(1 - 7*n*(7*n-3)*x/(1 + 9*x - ...

G.f.: 1/(1 + 16*x - 28*x/(1 - 21*x/(1 + 16*x - 154*x/(1 - 140*x/(1 + 16*x - ... - 7*n*(7*n-3)*x/(1 - 7*n*(7*n-4)*x/(1 + 16*x - .... (End)

Example

E.g.f.: A(x) = 1 + 12*x^2/2! + 732*x^4/4! + 109332*x^6/6! + 30406812*x^8/8! + 13587056052*x^10/10! + 8904250650492*x^12/12! +...
such that A(x) = (sin(3*x) + sin(4*x)) / sin(7*x).
O.g.f.: F(x) = 1 + 12*x + 732*x^2 + 109332*x^3 + 30406812*x^4 + 13587056052*x^5 + 8904250650492*x^6 + 8045727017033172*x^7 +...
such that the o.g.f. can be expressed as the continued fraction:
F(x) = 1/(1 - 3*4*x/(1 - 7^2*x/(1 - 10*11*x/(1 - 14^2*x/(1 - 17*18*x/(1 - 21^2*x/(1 - 24*25*x/(1 - 28^2*x/(1 - 31*32*x/(1 - 35^2*x/(1 - 38*39*x/(1 - ...)))))))))))).

Mathematica

With[{nn=40}, Take[CoefficientList[Series[(Sin[3x]+Sin[4x])/Sin[7x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Sep 23 2019 *)

Prog

(PARI) {a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(3*X) + sin(4*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(3*X) + cos(4*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(3*I*X) + exp(4*I*X))/(1 + exp(7*I*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A272158, A272467, A273031, A273032, A156196.

Sequence in context: A341560 A317337 A089036 * A276636 A284769 A201246

Adjacent sequences: A273030 A273031 A273032 * A273034 A273035 A273036

Keyword

nonn

Author

Paul D. Hanna, May 13 2016

Status

approved