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.

A295385 a(n) = n!*Sum_{k=0..n} binomial(2*n,n-k)*n^k/k!.

Original entry on oeis.org

1, 3, 32, 579, 14736, 483115, 19376928, 918980139, 50306339072, 3121729082739, 216541483852800, 16603614676249843, 1394473165806440448, 127308860552307549531, 12553171419275174137856, 1329537514269062031406875, 150531055969843353812533248, 18143286205523964035258551651
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2017

Keywords

Links

Crossrefs

Cf. A082545, A277373, A277423, A295384, A295406, A295407, A295408, A295409.

Programs

  • Magma
    [Factorial(n)*(&+[Binomial(2*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018
  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 17}]
Table[n! LaguerreL[n, n, -n], {n, 0, 17}]
Table[(-1)^n HypergeometricU[-n, n + 1, -n], {n, 0, 17}]
Join[{1}, Table[n! Sum[Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 17}]]
  • PARI
    for(n=0,30, print1(n!*sum(k=0,n, binomial(2*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

Formula

a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n+1).
a(n) = n!*Laguerre(n,n,-n).
a(n) ~ 2^(n - 1/2) * (1 + sqrt(2))^(n + 1/2) * n^n / exp((2 - sqrt(2))*n). - Vaclav Kotesovec, Nov 21 2017

A295408 a(n) = n! * Laguerre(n, 4*n, -n).

Original entry on oeis.org

1, 6, 134, 5052, 267576, 18246850, 1521907056, 150077897088, 17080661438336, 2203559337858174, 317761804144896000, 50650336389453807556, 8843008543955452118016, 1678231571506037926192698, 343989152383931539269349376, 75733086648535784012234565000
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 22 2017

Keywords

Comments

In general, for fixed m >= 1, n! * Sum_{k=0..n} binomial(m*n, n-k) * n^k / k! = n! * Laguerre(n, (m-1)*n, -n) ~ sqrt(1/2 + (m + 2)/(2*sqrt(m^2 + 4))) * (2^(m+1) * m^m / ((sqrt(m^2 + 4) - m) * (m - 2 + sqrt(m^2 + 4))^m))^n * exp((sqrt(m^2 + 4) - m)*n/2 - n) * n^n.

Links

Crossrefs

Cf. A277373 (m=1), A295385 (m=2), A295406 (m=3), A295407 (m=4).
Cf. A295409, A295418, A332679.

Programs

  • Magma
    [Factorial(n)*(&+[Binomial(5*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018
  • Mathematica
    Table[n!*LaguerreL[n,4*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[5*n,n-k]*n^k/k!,{k,0,n}],{n,1,15}]]
  • PARI
    for(n=0,30, print1(n!*sum(k=0, n, binomial(5*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018
  • PARI
    a(n) = n!*pollaguerre(n, 4*n, -n); \\ Michel Marcus, Feb 05 2021

Formula

a(n) = n!*Sum_{k=0..n} binomial(5*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 7/(2*sqrt(29))) * (131 - 22*sqrt(29))^n * exp((sqrt(29)-7)*n/2) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(4*n+1). - Ilya Gutkovskiy, Nov 23 2017

A295418 a(n) = n! * Laguerre(n, n*(n-1), -n).

Original entry on oeis.org

1, 2, 32, 1422, 124832, 18246850, 4005713952, 1232956594814, 506672220394496, 267992015325604578, 177340024595660672000, 143531889358151618790862, 139482579412432078779322368, 160267575964062522718064075618, 214924620455826226723051817295872
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 22 2017

Keywords

Links

Crossrefs

Cf. A277373, A295385, A295406, A295407, A295408, A295409.

Programs

  • Magma
    [Factorial(n)*(&+[Binomial(n^2, n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, May 13 2018
  • Mathematica
    Table[n!*LaguerreL[n,n*(n-1),-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[n^2,n-k]*n^k/k!,{k,0,n}],{n,1,15}]]
  • PARI
    for(n=0,25, print1(n!*sum(k=0,n, binomial(n^2, n-k)*n^k/k!), ", ")) \\ G. C. Greubel, May 13 2018
  • PARI
    a(n) = n!*pollaguerre(n, n*(n-1), -n); \\ Michel Marcus, Feb 05 2021

Formula

a(n) = n! * Sum_{k=0..n} binomial(n^2,n-k)*n^k/k!.
a(n) ~ exp(1/2) * n^(2*n).
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n^2-n+1). - Ilya Gutkovskiy, Nov 23 2017
Showing 1-3 of 3 results.

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