login
A289192
A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.
17
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
0,5
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
EXAMPLE
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, ...
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);
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 *)
PROG
(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
(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
CROSSREFS
Rows n=0-2 give: A000012, A000027(k+1), A008865(k+2).
Main diagonal gives A277373.
Sequence in context: A299500 A330141 A007441 * A111933 A144304 A122941
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 28 2017
STATUS
approved