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.

A128230 Expansion of exp(x)/(1 - x - x^2/2!), where a(n) = 1 + n*a(n-1) + C(n,2)*a(n-2).

Original entry on oeis.org

1, 2, 6, 25, 137, 936, 7672, 73361, 801705, 9856342, 134640146, 2023140417, 33163934641, 588936102860, 11263023492372, 230783643185881, 5044101110058737, 117136294344278346, 2880200768035996990
Offset: 0

Views

Author

Paul D. Hanna, Feb 20 2007

Keywords

Examples

			E.g.f.: exp(x)/(1 - x - x^2/2!) = 1 + 2*x + 6*x^2/2! + 25*x^3/3! + 137*x^4/4! + 936*x^5/5! + 7672*x^6/6! +... + a(n)*x^n/n! +...
where a(n) = 1 + n*a(n-1) + n*(n-1)*a(n-2)/2.

Links

Crossrefs

Cf. A087214, A000629, A128231, A128232.

Programs

  • Mathematica
    CoefficientList[Series[E^x/(1-x-x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)
  • PARI
    a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x-x^2/2! +x*O(x^n)),n)

Formula

a(n) ~ n!*exp(sqrt(3)-1)*((1+sqrt(3))/2)^(n+1)/sqrt(3) . - Vaclav Kotesovec, Oct 20 2012

A128232 Expansion of exp(x)/(1 - x^4/4!), where a(n) = 1 + C(n,4)*a(n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 141, 757, 3361, 11881, 69796, 541256, 3364362, 16217566, 127028721, 1288189281, 10294947721, 62859285817, 615454153246, 7709812846786, 75307542579116, 556618975909536, 6539815832391997
Offset: 0

Views

Author

Paul D. Hanna, Feb 20 2007

Keywords

Examples

			E.g.f.: exp(x)/(1 - x^4/4!) = 1 + x + 1*x^2/2! + 1*x^3/3! + 2*x^4/4! + 6*x^5/5! + 16*x^6/6! +... + a(n)*x^n/n! +...
where a(n) = 1 + n*(n-1)*(n-2)*(n-3)*a(n-4)/4!.

Crossrefs

Cf. A087214, A128230, A128231.

Programs

  • Maple
    G(x):=exp(x)/(1-x^4/4!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..24); # Zerinvary Lajos, Apr 03 2009
  • PARI
    a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x^4/4! +x*O(x^n)),n)
  • PARI
    /* Recurrence: */ a(n)=if(n<0,0,if(n<4,1,1 + n*(n-1)*(n-2)*(n-3)*a(n-4)/4!))
Showing 1-2 of 2 results.

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

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