OFFSET
0,3
LINKS
David Dickinson, On Lommel and Bessel polynomials, AMS Proceedings 1954.
Eric Weisstein's World of Mathematics, Lommel Polynomial.
FORMULA
T(n, k) = [z^k] 2*n*z*p(n-1, z) - p(n-2, z) where p(-1, z) = 0 and p(0, z) = 1.
T(n, k) = (-1)^k * [z^k] h(n, -n, z) where h(n, v, z) are the modified Lommel polynomials (A369117).
EXAMPLE
The list of coefficients starts:
[0] 1
[1] 0, 2
[2] -1, 0, 8
[3] 0, -8, 0, 48
[4] 1, 0, -72, 0, 384
[5] 0, 18, 0, -768, 0, 3840
[6] -1, 0, 288, 0, -9600, 0, 46080
[7] 0, -32, 0, 4800, 0, -138240, 0, 645120
[8] 1, 0, -800, 0, 86400, 0, -2257920, 0, 10321920
MAPLE
p := proc(n, x) option remember; if n = -1 then 0 elif n = 0 then 1 else
2*n*z*p(n - 1, z) - p(n - 2, z) fi end:
seq(seq(coeff(p(n, z), z, k), k = 0..n), n = 0..9);
MATHEMATICA
Table[CoefficientList[Expand[ResourceFunction["LommelR"][n, 1, 1/z]], z], {n, 0, 8}] // MatrixForm
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jan 30 2024
STATUS
approved