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.

A009123 Expansion of e.g.f.: cosh(log(1+sin(x))).

Original entry on oeis.org

1, 0, 1, -3, 8, -30, 136, -693, 3968, -25260, 176896, -1351383, 11184128, -99680490, 951878656, -9695756073, 104932671488, -1202439837720, 14544442556416, -185185594118763, 2475749026562048, -34674437196568950
Offset: 0

Views

Author

R. H. Hardin

Keywords

Comments

|a(n)| = number of even alternating permutations on n letters (offset 1). - Vladeta Jovovic, May 20 2007

Links

Crossrefs

Cf. A000111.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cosh(Log(1+Sin(x))))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 26 2018
  • Mathematica
    CoefficientList[Series[(1 + (1 + Sin[x])^2)/(2*(1 + Sin[x])), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 22 2015 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(cosh(log(1+sin(x))))) \\ G. C. Greubel, Jul 26 2018

Formula

a(2*n) = (1/2)*A000182(n+1); a(2*n+1) = A012007(n+1) = A009567(2*n+1) + 1.
G.f.: (1+x/(1+x^2))/2 + 1/2/Q(0) where Q(k) = 1 + (k+1)*x - x^2*(k+1)*(k+2)/2 /Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 12 2013
a(n) ~ n! * n * (-1)^n * (2/Pi)^(n+2). - Vaclav Kotesovec, Jan 22 2015

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

A009567 Expansion of e.g.f.: sinh(log(1 + sin(x))).

Original entry on oeis.org

0, 1, -1, 2, -8, 31, -136, 692, -3968, 25261, -176896, 1351382, -11184128, 99680491, -951878656, 9695756072, -104932671488, 1202439837721, -14544442556416, 185185594118762, -2475749026562048, 34674437196568951
Offset: 0

Views

Author

R. H. Hardin

Keywords

Comments

|a(n)| = number of odd alternating permutations on n letters (offset 1). - Vladeta Jovovic, May 20 2007

Links

Crossrefs

Cf. A000111, A009123.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Sinh[Log[1+Sin[x]]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 11 2014 *)
  • PARI
    x='x+O('x^30); concat([0], Vec(serlaplace(sinh(log(1+sin(x)))))) \\ G. C. Greubel, Jan 21 2018

Formula

a(2*n) = A000182(n+1)/2.
a(2*n+1) = A012007(n+1) - 1.

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Definition clarified and previous Mathematica program replaced by Harvey P. Dale, May 11 2014

A029584 Expansion of cos x + tan x + sec x.

Original entry on oeis.org

2, 1, 0, 2, 6, 16, 60, 272, 1386, 7936, 50520, 353792, 2702766, 22368256, 199360980, 1903757312, 19391512146, 209865342976, 2404879675440, 29088885112832, 370371188237526, 4951498053124096, 69348874393137900
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(2n) = 2*A012007(n), a(2n+1) = A000182(n+1).

Links

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Cos[x]+Tan[x]+Sec[x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 09 2025 *)

A012009 Expansion of e.g.f. sech(log(cos(x))) (even exponents only).

Original entry on oeis.org

1, 0, -3, -30, -63, 12540, 602877, 6625710, -1991169183, -241970036520, -7540177734243, 2917041754949850, 699983161534169697, 46722975483964508820, -21334067257986056115363, -8882421213380429461235610, -1081286159351846822872767423
Offset: 0

Views

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

Examples

			1 - (3/4!)*x^4 - (30/6!)*x^6 - (63/8!)*x^8 + (12540/10!)*x^10 ...

Crossrefs

Cf. A012003, A012007, A012008.

Programs

  • Mathematica
    With[{nn=40},Take[CoefficientList[Series[Sech[Log[Cos[x]]],{x,0,nn}], x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Dec 16 2011 *)

Formula

E.g.f.: sech(log(cos(x))) = 2*cos(x)/(1+cos^2(x)) is the reciprocal of the e.g.f. of A012007. - Peter Bala, Dec 02 2011

Extensions

More terms from Harvey P. Dale, Dec 16 2011
New name from Michel Marcus, Aug 04 2024
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.