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 1-4 of 4 results.

A002437 a(n) = A000364(n) * (3^(2*n+1) + 1)/4.

Original entry on oeis.org

1, 7, 305, 33367, 6815585, 2237423527, 1077270776465, 715153093789687, 626055764653322945, 698774745485355051847, 968553361387420436695025, 1632180870878422847476890007, 3286322019402928956112227932705, 7791592461957309952817483706344167, 21485762937086358457367440231243675985
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The terms are multiples of the Euler numbers (A000364).

Examples

			a(4) = A000364(4) * (3^(2*4+1)+1)/4 = 1385 * (3^9+1)/4 = 1385 * 4921 = 6815585.

References

  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 75.
  • J. W. L. Glaisher, Messenger of Math., 28 (1898), 36-79, see esp. p. 51.
  • L. B. W. Jolley, Summation of Series, Dover, 2nd ed. (1961)
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Bisections: A156168, A156169.
Cf. A000191, A001209, A012494, A000364, A000281, A156134, A349429.
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), A156134 (k=3).

Programs

  • Maple
    Q:=proc(n) option remember; if n=0 then RETURN(1); else RETURN(expand((u^2+1)*diff(Q(n-1),u)+u*Q(n-1))); fi; end;
[seq(subs(u=sqrt(3),Q(2*n)),n=0..25)];
  • Mathematica
    Table[Abs[EulerE[2 n]] (3^(2 n + 1) + 1) / 4, {n, 0, 30}] (* Vincenzo Librandi, Feb 07 2017 *)

Formula

A000364(n) * (3^(2*n+1) + 1)/4.
Q_2n(sqrt(3)), where the polynomials Q_n() are defined in A104035. - N. J. A. Sloane, Nov 06 2009
a(n) = (-1)^n*Sum_{k = 0..2*n-1} w^(2*n+k)*Sum_{j = 1..2*n-1} (-1)^(k-j)*binomial(2*n-1,k-j)*(2*j - 1)^(2*n-2), where w = exp(2*Pi*i/6) (i = sqrt(-1)). Cf. A002439. - Peter Bala, Jan 21 2011
Sum_{n>=1} (-1)^floor((n-1)/2) 1/A007310(n)^s = r_s with r_{2s+1} = 2 *(Pi/6)^(2s+1) *a(s) /(2s)!. [Jolley eq (315)]. - R. J. Mathar, Mar 24 2011
From Peter Bala, Feb 06 2017: (Start)
E.g.f.: cos(x)^2/cos(3*x) = cos(x)/(1 - 4*sin(x)^2) = 1 + 7*x^2/2! + 305*x^4/4! + 33367*x^6/6! + .... This is the even part of (1/2)*sec(x + Pi/3). Cf. A000191. (End)
a(n) = (1/2)*Integral_{x = 0..inf} x^(2*n)*cosh(Pi*x/3)/cosh(Pi*x/2) dx. - Cf. A000281. - Peter Bala, Nov 08 2019

Extensions

More terms from Herman P. Robinson
Further terms from N. J. A. Sloane, Nov 06 2009

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

Views

Author

N. J. A. Sloane, Nov 06 2009

Keywords

Links

Crossrefs

Cf. A012494, A001209, A000364, A000281, A002437.
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

A296730 Expansion of e.g.f. arctanh(x*cos(x)) (odd powers only).

Original entry on oeis.org

1, -1, -31, -337, 24705, 2451679, -17936543, -42895630065, -5396647099903, 1239561882325439, 708575518706816481, 37448619025871342959, -113842057082636742446975, -52054011876398495316250977, 16226448322449614832534708065, 31975745831751940004484917311439
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arctanh(x*cos(x)) = x/1! - x^3/3! - 31*x^5/5! - 337*x^7/7! + 24705*x^9/9! + ...

Crossrefs

Cf. A000364, A009015, A009016, A009446, A009447, A009633, A009634, A010050, A012494, A191512, A296465, A296467, A296728, A296729, A296731, A296740.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcTanh[x Cos[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(atanh(x*cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arctanh(x*cos(x)).

A101923 Expansion of 2 * arccot(cos(x)).

Original entry on oeis.org

1, 2, 1, -148, -3719, -20098, 5055961, 403644152, 7831409041, -2707151879398, -472143935754479, -22085804322342748, 9362259685093715401, 2995219209329323622102, 274269338931958691728681, -132963342779629343323496848, -70698673853383423350187244639
Offset: 1

Views

Author

Ralf Stephan, Dec 27 2004

Keywords

Comments

Odd coefficients are zero.

Links

Crossrefs

Cf. A012494, A000364, A000464, A156138, A002439.
Cf. other sequences with a g.f. of the form sin(x)/(1 - k*sin^2(x)): A012494 (k=-1), A000364 (k=1), A000464 (k=2), A156138 (k=3), A002439 (k=4).

Programs

  • Maple
    with(gfun):
series(sin(x)/(1-(1/2)*sin(x)^2), x, 35):
L := seriestolist(%):
seq(op(2*i, L)*(2*i-1)!, i = 1..floor((1/2)*nops(L)));
# Peter Bala, Feb 06 2017
  • Mathematica
    With[{nn=40},Take[CoefficientList[Series[2ArcCot[Cos[x]],{x,0,nn}],x] Range[0,nn]!,{3,-1,2}]] (* Harvey P. Dale, Nov 17 2014 *) (* adapted by Vincenzo Librandi, Feb 07 2017 *)

Formula

2*acot(cos(x)) = Pi/2 + x^2/2! + 2*x^4/4! + x^6/6! - 148*x^8/8! - 3719*x^10/10! -...
2*atan(cos(x)) = Pi/2 - x^2/2! - 2*x^4/4! - x^6/6! + 148*x^8/8! + 3719*x^10/10! +...
G.f. sin(x)/(1 - 1/2*sin(x)^2) = x + 2*x^3/3! + x^5/5! - 148*x^7/7! - ... - Peter Bala, Feb 06 2017

Extensions

More terms from Harvey P. Dale, Nov 17 2014
Signs of the data entries corrected by Peter Bala, Feb 06 2017
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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