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 1-5 of 5 results.

A296842 Expansion of e.g.f. cos(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, -3, -15, -14, 1755, 60357, 1740284, 45816165, 776485557, -37342503290, -7203185712261, -822818831400759, -85463040449605000, -8640073895507612019, -843669753827174738535, -73050419139737972150438, -3478007209663880122501701
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			cos(x*tan(x/2)) = 1 - 3*x^4/4! - 15*x^6/6! - 14*x^8/8! + 1755*x^10/10! + 60357*x^12/12! + ...

Crossrefs

Cf. A001469, A003710, A009080, A110501, A296839, A296841.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[Cos[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] cos(x*tan(x/2)).

A003711 Expansion of e.g.f. cos(tanh(x)) (even powers only).

Original entry on oeis.org

1, -1, 9, -177, 6097, -325249, 24807321, -2558036145, 342232522657, -57569080467073, 11879658510739497, -2948163649552594737, 865683568087537789297, -296699416391356495667713, 117330699580950022391960505
Offset: 0

Views

Author

R. H. Hardin, Simon Plouffe

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003710.

Programs

  • Mathematica
    nn = 20; Table[(CoefficientList[Series[Cos[Tanh[x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Feb 16 2015 *)
  • Maxima
    a(n):=sum((sum(binomial(2*m+k-1,2*m-1)*(2*m+k)!*(-1)^(k)*2^(2*n-2*m-k)*stirling2(2*n,2*m+k),k,0,2*n-2*m))/(2*m)!,m,1,n); /* Vladimir Kruchinin, Jun 10 2011 */

Formula

a(n) = Sum_{m=1..n} ( Sum_{k=0..2*n-2*m} binomial(2*m+k-1,2*m-1) * (2*m+k)! * (-1)^k * 2^(2*n-2*m-k) * Stirling2(2*n,2*m+k) )/(2*m)!, n>0, a(0)=1. - Vladimir Kruchinin, Jun 10 2011

A003723 E.g.f. exp(tanh(x)).

Original entry on oeis.org

1, 1, 1, -1, -7, -3, 97, 275, -2063, -15015, 53409, 968167, -752343, -77000363, -166831871, 7433044411, 43685848289, -843598411471, -9398558916159, 107426835190735, 2116926930779225, -14072980460605907
Offset: 0

Views

Author

R. H. Hardin, Simon Plouffe

Keywords

Comments

Row sums of triangle A111593.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003706, A003710.

Programs

  • Mathematica
    With[{nn = 30}, CoefficientList[Series[Exp[Tanh[x]], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Apr 11 2014 *)
  • Maxima
    a(n):=if n=0 then 1 else sum(sum(binomial(k-1,m-1)*k!*(-1)^(m+k)*2^(n-k)*stirling2(n,k),k,m,n)/m!,m,1,n); /* Vladimir Kruchinin, Jun 28 2011 */

Formula

a(n) := sum(m=1..n, sum(k=m..n, binomial(k-1,m-1)*k!*(-1)^(m+k)*2^(n-k)*Stirling2(n,k))/m!), n>0, a(0)=1. - Vladimir Kruchinin, Jun 28 2011

A013521 Numerator of [x^(2n)] in the Taylor expansion cos(cosec(x)-cot(x))= 1-x^2/8 -7*x^4/384 -97*x^6/46080 -2063*x^8/10321920 -17803*x^10/1238630400 -....

Original entry on oeis.org

1, -1, -7, -97, -2063, -17803, -250781, 166831871, 43685848289, 447550424579, 84677077231169, 11657476758734011, 28924058075775365981, 44287070229737735633567, 305190813989360271816409
Offset: 0

Views

Author

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

Keywords

Comments

The e.g.f of x/2, cos(cosec(x)-cot(x)) = 1 -1*x^2/(2^2*2!) -7*x^4/(2^4*4!) -97*x^6/(2^6*6!) -2063*x^8/(2^8*8!) -..., is apparently covered by A003710.

Links

Programs

  • Mathematica
    Numerator[Take[CoefficientList[Series[Cos[Csc[x] - Cot[x]], {x, 0, 25}], x], {1, -1, 2}]] (* G. C. Greubel, Nov 12 2016 *)

Extensions

Name edited by R. J. Mathar, Dec 19 2011

A297215 Expansion of e.g.f. exp(cos(tan(x))-1) (even powers only).

Original entry on oeis.org

1, -1, -4, -7, 1003, 64836, 3350349, 104475395, -12291888052, -4268687337603, -877769324284177, -139938933307889412, -9581950082738688167, 6333750977985105075527, 4837035706491587870342140, 2439859866050865745230242689, 1033093869484852949078289394195
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			exp(cos(tan(x))-1) = 1 - x^2/2! - 4*x^4/4! - 7*x^6/6! + 1003*x^8/8! + 64836*x^10/10! + ...

Crossrefs

Cf. A003710, A009074, A009201, A009202, A009203, A009204, A009238, A009239, A009240, A009241, A009254, A297214.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Exp[Cos[Tan[x]] - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(cos(tan(x))-1).
Showing 1-5 of 5 results.

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