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.

Previous Showing 11-12 of 12 results.

A156134 Q_2n(sqrt(2)) (see A104035).

Original entry on oeis.org

1, 5, 157, 12425, 1836697, 436366445, 152053957237, 73053601590065, 46283414838553777, 37386890114969267285, 37503815980582784378317, 45739346519434253222582105, 66650214918099514832427062857, 114363498315755726948758209518525, 228234739109951323288351261455519397
Offset: 0

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Author

N. J. A. Sloane, Nov 06 2009

Keywords

Crossrefs

Cf. other sequences with a g.f. of the form cos(x)/(1 - k*sin^2(x)): A012494 (k=-1), A001209 (k=1/2), A000364(k=1), A000281 (k=2), A002437 (k=4).

Programs

  • Maple
    with(gfun):
    series(cos(x)/(1-3*sin(x)^2), x, 30):
    L := seriestolist(%):
    seq(op(2*i-1,L)*(2*i-2)!, i = 1..floor((1/2)*nops(L)));
    # Peter Bala, Feb 06 2017
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Cos[x]/(1 - 3*Sin[x]^2), {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]] (* G. C. Greubel, Mar 29 2018 *)
  • PARI
    x='x+O('x^50); v=Vec(serlaplace(cos(x)/(1 - 3*sin(x)^2))); vector(#v\2,n,v[2*n-1]) \\ G. C. Greubel, Mar 29 2018

Formula

G.f. cos(x)/(1 - 3*sin(x)^2) = 1 + 5*x^2/2! + 157*x^4/4! + 12425*x^6/6! + ... - Peter Bala, Feb 06 2017

A352977 Expansion of e.g.f. cos(2x) cos(3x) / cos(6x) (even powers only).

Original entry on oeis.org

1, 23, 3985, 1743623, 1424614945, 1870693029623, 3602792061891505, 9566946196183630823, 33500193836861731481665, 149565522713623779723211223, 829235405016410370201483113425, 5589623533324449496004527793434823, 45017811997394066193946619670380594785
Offset: 0

Views

Author

F. Chapoton, Apr 13 2022

Keywords

Comments

Only terms of even index are given. Terms of odd index are zero.

Crossrefs

Intermediate case between A002437 and A349429.
Cf. A000192.

Programs

  • Maple
    egf := (cos(x) + cos(5*x))*sec(6*x) / 2: ser := series(egf, x, 32):
    seq(n!*coeff(ser, x ,n), n = 0..24, 2); # Peter Luschny, Apr 13 2022
  • PARI
    my(x='x+O('x^30)); select(x->(x>0), Vec(serlaplace(cos(2*x)*cos(3*x)/cos(6*x)))) \\ Michel Marcus, Apr 13 2022
  • Sage
    x = PowerSeriesRing(QQ, 'x', default_prec=30).gen()
    f = cos(2*x) * cos(3*x) / cos(6*x)
    [cf for cf in f.egf_to_ogf() if cf]
    

Formula

E.g.f.: cos(2*x) * cos(3*x) / cos(6*x).
From Peter Luschny, Apr 13 2022: (Start)
E.g.f.: (cos(x) + cos(5*x))*sec(6*x) / 2, even powers only.
a(n) = A000192(n)/2. (End)
a(n) ~ 2^(6*n + 3/2) * 3^(2*n + 1/2) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 15 2022
Previous Showing 11-12 of 12 results.