A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 14, 34, 24, 1, 5, 23, 86, 209, 120, 1, 6, 34, 168, 648, 1546, 720, 1, 7, 47, 286, 1473, 5752, 13327, 5040, 1, 8, 62, 446, 2840, 14988, 58576, 130922, 40320, 1, 9, 79, 654, 4929, 32344, 173007, 671568, 1441729, 362880
Offset: 0
Examples
Square array A(n,k) begins: : 1, 1, 1, 1, 1, 1, ... : 1, 2, 3, 4, 5, 6, ... : 2, 7, 14, 23, 34, 47, ... : 6, 34, 86, 168, 286, 446, ... : 24, 209, 648, 1473, 2840, 4929, ... : 120, 1546, 5752, 14988, 32344, 61870, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
- Eric Weisstein's World of Mathematics, Laguerre Polynomial
- Wikipedia, Laguerre polynomials
- Index entries for sequences related to Laguerre polynomials
Crossrefs
Programs
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Maple
A:= (n,k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n): seq(seq(A(n, d-n), n=0..d), d=0..12);
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Mathematica
A[n_, k_] := n! * LaguerreL[n, -k]; Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *) -
PARI
{T(n, k) = if(n<2, k*n+1, (2*n+k-1)*T(n-1, k)-(n-1)^2*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021 -
PARI
T(n, k) = n!*pollaguerre(n, 0, -k); \\ Michel Marcus, Feb 05 2021
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Python
from sympy import binomial, factorial as f def A(n, k): return f(n)*sum(binomial(n, i)*k**i/f(i) for i in range(n + 1)) for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Jun 28 2017
Formula
A(n,k) = n! * Sum_{i=0..n} k^i/i! * binomial(n,i).
E.g.f. of column k: exp(k*x/(1-x))/(1-x).
A(n, k) = (-1)^n*KummerU(-n, 1, -k). - Peter Luschny, Feb 12 2020
A(n, k) = (2*n+k-1)*A(n-1, k) - (n-1)^2*A(n-2, k) for n > 1. - Seiichi Manyama, Feb 03 2021