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.

A317279 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*n^k*n!/k!.

Original entry on oeis.org

1, 1, 0, -9, -32, 225, 3456, 2695, -433152, -4495743, 47872000, 1768142871, 6703534080, -597265448351, -11959736205312, 126058380654375, 9454322092343296, 84694164336894465, -5776865438988238848, -192541299662555831753, 1511905067561779200000, 243338391925401706938081, 3972949090873574466519040
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 25 2018

Keywords

Comments

a(n) is the n-th term of the inverse Lah transform of the powers of n.

Links

Crossrefs

Cf. A111884, A293145, A317277, A317278.

Programs

  • Magma
    l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
[1]cat[(-1)^(n+1)*Factorial(n)*l(n-1,1,n): n in [1..30]]; // G. C. Greubel, Mar 09 2021
  • Maple
    A317279:= n -> `if`(n=0,1,(-1)^(n+1)*n!*simplify(LaguerreL(n-1,1,n), 'LaguerreL'));
seq(A317279(n), n = 0..30); # G. C. Greubel, Mar 09 2021
  • Mathematica
    Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] n^k n!/k!, {k, n}], {n, 22}]]
Table[n! SeriesCoefficient[Exp[n x/(1 + x)], {x, 0, n}], {n, 0, 22}]
Table[n! SeriesCoefficient[Product[Exp[-n (-x)^k], {k, n}], {x, 0, n}], {n, 0, 22}]
Join[{1}, Table[(-1)^(n+1) n n! Hypergeometric1F1[1-n, 2, n], {n, 22}]]
  • PARI
    a(n) = if (n==0, 1, (-1)^(n+1)*n!*pollaguerre(n-1, 1, n)); \\ Michel Marcus, Mar 10 2021
  • Sage
    [1]+[(-1)^(n+1)*factorial(n)*gen_laguerre(n-1,1,n) for n in (1..30)] # G. C. Greubel, Mar 09 2021

Formula

a(n) = n! * [x^n] exp(n*x/(1 + x)).
a(n) = n! * [x^n] Product_{k>=1} exp(-n*(-x)^k).
a(n) = (-1)^(n+1) * n * n! * Hypergeometric1F1([1-n], [2], n) with a(0) = 1.
a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1, n) with a(0) = 1. - G. C. Greubel, Mar 09 2021

A317278 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*k^n*n!/k!.

Original entry on oeis.org

1, 1, 2, -15, -164, 4245, 46386, -4901939, 39141656, 11707820361, -671114863610, -29398709945319, 7385525824325364, -307076643365636963, -73748845974115224262, 14299745046516639280005, -237996466462017367478864, -377740669670216316717155055, 75515477307532501838072029326
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 25 2018

Keywords

Comments

a(n) is the n-th term of the inverse Lah transform of the n-th powers.

Links

Crossrefs

Cf. A111884, A256467, A317277, A317279.

Programs

  • Magma
    [1]cat[(&+[(-1)^(n+j)*Binomial(n-1,j-1)*Binomial(n,j)*Factorial(n-j)*j^n: j in [0..n]]): n in [1..30]]; // G. C. Greubel, Mar 09 2021
  • Maple
    A317278:= n-> `if`(n=0,1, add((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n));
seq(A317278(n), n=0..30); # G. C. Greubel, Mar 09 2021
  • Mathematica
    Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] k^n n!/k!, {k, n}], {n, 18}]]
Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 + x))^k/k!, {k, n}], {x, 0, n}], {n, 18}]]
  • PARI
    a(n) = if (n==0, 1, sum(k=0, n, (-1)^(n-k)*binomial(n-1, k-1)*k^n*n!/k!)); \\ Michel Marcus, Mar 10 2021; corrected Jun 13 2022
  • Sage
    [1]+[sum((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*factorial(n-j)*j^n for j in (0..n)) for n in (1..30)] # G. C. Greubel, Mar 09 2021

Formula

a(n) = n! * [x^n] Sum_{k>=0} k^n*(x/(1 + x))^k/k!.
Showing 1-2 of 2 results.

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