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 1-2 of 2 results.

A327034 Expansion of e.g.f. exp(x) / (2 - cosh(x)).

Original entry on oeis.org

1, 1, 2, 4, 14, 46, 242, 1114, 7814, 46246, 405482, 2933074, 30860414, 263817646, 3238391522, 31943268634, 448122565814, 5009616448246, 79063212894362, 987840438629794, 17322647732052014, 239217148602642046, 4614370558369770002, 69790939492563608554
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2019

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[x]/(2 - Cosh[x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * binomial(n,k) * (2^(k - 1) - 2) * a(n-k).
a(n) ~ n! * (7 + 4*sqrt(3) + (-1)^n) / ((3+2*sqrt(3)) * (log(2+sqrt(3)))^(n+1)). - Vaclav Kotesovec, Dec 03 2019

A081558 Exponential generating function: exp(cosh(x)+2*x-1).

Original entry on oeis.org

1, 2, 5, 14, 44, 152, 575, 2354, 10379, 48902, 245240, 1301984, 7294589, 42959282, 265263185, 1712168654, 11528506124, 80783015192, 588097479635, 4439382164114, 34699233959759, 280381494540182, 2339287666524440, 20125268756209664, 178348602246900569
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Comments

Old definition was "Second binomial transform of expansion of exp(cosh(x))".
Binomial transform of A081557.

Crossrefs

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(Cosh(x)+2*x-1) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 13 2019
    
  • Maple
    seq(coeff(series(exp(cosh(x)+2*x-1), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019
  • Mathematica
    With[{nn = 30}, CoefficientList[Series[Exp[Cosh[x] + 2 x - 1], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Aug 08 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(cosh(x)+2*x-1) )) \\ G. C. Greubel, Aug 13 2019
    
  • Sage
    [factorial(n)*( exp(cosh(x)+2*x-1) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Aug 13 2019

Formula

E.g.f.: exp(2*x) * exp(cosh(x))/e.

Extensions

Definition edited by N. J. A. Sloane, Dec 12 2021
Showing 1-2 of 2 results.