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 1-3 of 3 results.

A082308 Expansion of e.g.f. (1+x)*exp(4*x)*cosh(x).

Original entry on oeis.org

1, 5, 25, 127, 657, 3449, 18281, 97395, 519841, 2773741, 14776377, 78538343, 416367665, 2201517153, 11610231433, 61078202971, 320570884929, 1678897264085, 8775159682649, 45780628812879, 238431945108433
Offset: 0

Views

Author

Paul Barry, Apr 09 2003

Keywords

Comments

Binomial transform of A082307.

Links

Crossrefs

Cf. A082309.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(4*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[4*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 16 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace((1+x)*exp(4*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

Formula

a(n) = (A081105(n) + A006234(n))/2.
a(n) = ((n+3)*3^(n-1) + (n+5)*5^(n-1))/2.
G.f.: ((1-4*x)/(1-5*x)^2 + (1-2*x)/(1-3*x)^2)/2.
E.g.f.: (1+x)*exp(4*x)*cosh(x) = (1+x)*(exp(5*x) + exp(3*x))/2.

A082306 Expansion of e.g.f. (1+x)*exp(2*x)*cosh(x).

Original entry on oeis.org

1, 3, 9, 29, 97, 327, 1097, 3649, 12033, 39371, 127945, 413349, 1328609, 4251535, 13551753, 43046729, 136314625, 430467219, 1355971721, 4261625389, 13366006881, 41841412823, 130754415049, 407953774929, 1270932914177
Offset: 0

Views

Author

Paul Barry, Apr 09 2003

Keywords

Comments

Binomial transform of A082305 a(n)=(A006234(n)+A000027(n))/2

Links

Crossrefs

Cf. A082307.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(2*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[2*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 16 2018 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace((1+x)*exp(2*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

Formula

a(n) = (n + 1 + 3^(n-1)*(n + 3))/2.
G.f.: (1/(1-x)^2 + (1-2*x)/(1-3*x)^2)/2.
E.g.f.: (1+x)*exp(2*x)*cosh(x).

A082309 Expansion of e.g.f.: (1+x)*exp(5*x)*cosh(x).

Original entry on oeis.org

1, 6, 36, 218, 1336, 8280, 51776, 325792, 2057856, 13023104, 82456576, 521826816, 3298727936, 20822038528, 131210919936, 825373859840, 5182772248576, 32487861092352, 203308891897856, 1270289732337664, 7924975155019776
Offset: 0

Views

Author

Paul Barry, Apr 09 2003

Keywords

Comments

Binomial transform of A082307.

Links

Crossrefs

Cf. A082308, A082307, A082306, A082305.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(5*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018
  • Mathematica
    With[{nn=30},CoefficientList[Series[(1+x)Exp[5x]Cosh[x],{x,0,nn}],x]Range[0,nn]!] (* or *) LinearRecurrence[{20,-148,480,-576},{1,6,36,218},30] (* Harvey P. Dale, Aug 27 2012 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace((1+x)*exp(5*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

Formula

a(n) = (A081106(n) + A079028(n))/2.
a(n) = ((n+4)*4^(n-1) + (n+6)*6^(n-1))/2.
G.f.: ((1-5*x)/(1-6*x)^2 + (1-3*x)/(1-4*x)^2)/2.
From Harvey P. Dale, Aug 27 2012: (Start)
E.g.f.: (1+x)*exp(5*x)*cosh(x).
a(n) = 20*a(n-1) - 148*a(n-2) + 480*a(n-3) - 576*a(n-4), n>3. (End)

Extensions

Definition clarified by Harvey P. Dale, Aug 27 2012
Showing 1-3 of 3 results.

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