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.

A080527 Expansion of e.g.f. exp(3*cosh(x))/e^3 (even powers only).

Original entry on oeis.org

1, 3, 30, 543, 14745, 546618, 26119965, 1547467743, 110507211390, 9310580616243, 910032696123645, 101790848712790218, 12883623878563854105, 1827803943114479006043, 288318381606931126782270, 50215020818975432279332743, 9597691024295026236008687265
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Comments

Periodic zeros suppressed.

Examples

			exp(3*cosh(x))/exp(3) = 1 + 3*x^2/2! + 30*x^4/4! + ...
		

Crossrefs

Programs

  • Mathematica
    With[{nn=30},Take[CoefficientList[Series[Exp[3Cosh[x]]/E^3,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Dec 15 2013 *)

Formula

E.g.f.: exp(3*cosh(x)-3).
a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(2*n-1,2*k-1) * a(n-k). - Ilya Gutkovskiy, Mar 10 2022

A081566 Second binomial transform of expansion of exp(3cosh(x)).

Original entry on oeis.org

1, 2, 7, 26, 118, 572, 3127, 18146, 114793, 765602, 5463982, 40870436, 323326813, 2667777842, 23092966267, 207651618746, 1947316349278, 18906249136892, 190564801592107, 1982986181092226, 21345005629846213, 236628248493001202
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Comments

Binomial transform of A081565.

Crossrefs

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(3*Cosh(x)+2*x-3) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 13 2019
    
  • Maple
    seq(coeff(series(exp(3*cosh(x)+2*x-3), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019
  • Mathematica
    With[{nn = 30}, CoefficientList[Series[Exp[3 Cosh[x] + 2 x - 3], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Aug 08 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(3*cosh(x)+2*x-3) )) \\ G. C. Greubel, Aug 13 2019
    
  • Sage
    [factorial(n)*( exp(3*cosh(x)+2*x-3) ).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(3*cosh(x))/e^3 = exp(3*cosh(x)+2*x-3).
Showing 1-2 of 2 results.