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.

A195254 O.g.f.: Sum_{n>=0} 2*(n+2)^(n-1)*x^n/(1+n*x)^n.

Original entry on oeis.org

1, 2, 6, 20, 76, 336, 1744, 10592, 74400, 595712, 5362432, 53626368, 589894144, 7078737920, 92023609344, 1288330563584, 19324958519296, 309199336439808, 5256388719738880, 94614996955824128, 1797684942161707008, 35953698843236237312, 755027675707965177856
Offset: 0

Views

Author

Paul D. Hanna, Sep 13 2011

Keywords

Comments

Compare the g.f. to: W(x)^2 = Sum_{n>=0} 2*(n+2)^(n-1)*x^n/n! where W(x) = LambertW(-x)/(-x).
Compare to a g.f. of A000522: Sum_{n>=0} (n+1)^(n-1)*x^n/(1+n*x)^n, which generates the total number of arrangements of a set with n elements.

Examples

			O.g.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 336*x^5 + 1744*x^6 +...
where
A(x) = 1 + 2*x/(1+x) + 2*4*x^2/(1+2*x)^2 + 2*5^2*x^3/(1+3*x)^3 + 2*6^3*x^4/(1+4*x)^4 +...

Links

Crossrefs

Cf. A000522, A195255, A195256, A195257.

Programs

  • Mathematica
    Flatten[{1,Table[(n-1)!*Sum[2^k/(k-1)!,{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 17 2013 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,2*(m+2)^(m-1)*x^m/(1+m*x+x*O(x^n))^m),n)}
  • PARI
    {a(n)=if(n==0,1,(n-1)!*sum(k=1,n,2^k/(k-1)!))}

Formula

a(n) = (n-1)!*Sum_{k=1..n} 2^k/(k-1)! for n>0, with a(0)=1.
Recurrence: a(n) = (n+1)*a(n-1) - 2*(n-2)*a(n-2). - Vaclav Kotesovec, Aug 17 2013
a(n) ~ 2*exp(2) * (n-1)!. - Vaclav Kotesovec, Aug 17 2013

A195255 O.g.f.: Sum_{n>=0} 3*(n+3)^(n-1)*x^n/(1+n*x)^n.

Original entry on oeis.org

1, 3, 12, 51, 234, 1179, 6624, 41931, 300078, 2420307, 21841812, 218595267, 2405079378, 28862546859, 375217892136, 5253064838811, 78796015628886, 1260736379202339, 21432518833860252, 385785340171746003, 7329921466749958458, 146598429345459522363
Offset: 0

Views

Author

Paul D. Hanna, Sep 13 2011

Keywords

Comments

Compare the g.f. to: W(x)^3 = Sum_{n>=0} 3*(n+3)^(n-1)*x^n/n! where W(x) = LambertW(-x)/(-x).
Compare to a g.f. of A000522: Sum_{n>=0} (n+1)^(n-1)*x^n/(1+n*x)^n, which generates the total number of arrangements of a set with n elements.

Examples

			O.g.f.: A(x) = 1 + 3*x + 12*x^2 + 51*x^3 + 234*x^4 + 1179*x^5 +...
where
A(x) = 1 + 3*x/(1+x) + 3*5*x^2/(1+2*x)^2 + 3*6^2*x^3/(1+3*x)^3 + 3*7^3*x^4/(1+4*x)^4 +..

Crossrefs

Cf. A000522, A195254, A195256, A195257.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0,n,3*(m+3)^(m-1)*x^m/(1+m*x+x*O(x^n))^m),n)}
  • PARI
    {a(n)=if(n==0,1,(n-1)!*sum(k=1,n,3^k/(k-1)!))}

Formula

a(n) = (n-1)!*Sum_{k=1..n} 3^k/(k-1)! for n>0, with a(0)=1.
a(n) ~ 3*exp(3) * (n-1)!. - Vaclav Kotesovec, Oct 10 2020

A195256 O.g.f.: Sum_{n>=0} 4*(n+4)^(n-1)*x^n/(1+n*x)^n.

Original entry on oeis.org

1, 4, 20, 104, 568, 3296, 20576, 139840, 1044416, 8617472, 78605824, 790252544, 8709555200, 104581771264, 1359831461888, 19038714208256, 285585008091136, 4569377309327360, 77679482978041856, 1398230968482660352, 26566389500682174464
Offset: 0

Views

Author

Paul D. Hanna, Sep 13 2011

Keywords

Comments

Compare the g.f. to: W(x)^4 = Sum_{n>=0} 4*(n+4)^(n-1)*x^n/n! where W(x) = LambertW(-x)/(-x).
Compare to a g.f. of A000522: Sum_{n>=0} (n+1)^(n-1)*x^n/(1+n*x)^n, which generates the total number of arrangements of a set with n elements.

Examples

			O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 104*x^3 + 568*x^4 + 3296*x^5 +...
where
A(x) = 1 + 4*x/(1+x) + 4*6*x^2/(1+2*x)^2 + 4*7^2*x^3/(1+3*x)^3 + 4*8^3*x^4/(1+4*x)^4 +...

Crossrefs

Cf. A000522, A195254, A195255, A195257.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0,n,4*(m+4)^(m-1)*x^m/(1+m*x+x*O(x^n))^m),n)}
  • PARI
    {a(n)=if(n==0,1,(n-1)!*sum(k=1,n,4^k/(k-1)!))}

Formula

a(n) = (n-1)!*Sum_{k=1..n} 4^k/(k-1)! for n>0, with a(0)=1.
a(n) ~ 4*exp(4) * (n-1)!. - Vaclav Kotesovec, Oct 10 2020
Showing 1-3 of 3 results.

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