cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

1, 6, 330, 48726, 13534410, 6046913046, 3962771924490, 3580686141374166, 4266519857080266570, 6481738795978992136086, 12228451239686387772736650, 28048508112504152087554462806, 76867928701091608252297826870730, 248058932215537567368765344245378326, 931049990613171839116868739409352364810, 4021504762182514582910341826029900914866646
Offset: 0

Views

Author

Paul D. Hanna, May 13 2016

Keywords

Examples

			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 - ...)))))))))))).

Crossrefs

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

Programs

  • 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 *)
  • 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), ", "))

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

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

Original entry on oeis.org

1, 12, 732, 109332, 30406812, 13587056052, 8904250650492, 8045727017033172, 9586782871360007772, 14564334832981893064692, 27477080512619965247054652, 63024425641459625896776174612, 172720667970739808701108304367132, 557383361208023769780400587942586932, 2092050338949043346342979863638489321212, 9036239176876728629700436615577988154925652
Offset: 0

Views

Author

Paul D. Hanna, May 13 2016

Keywords

Examples

			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 - ...)))))))))))).

Links

Crossrefs

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

Programs

  • 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 *)
  • 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), ", "))

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)
Showing 1-2 of 2 results.

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