A369585 Table read by rows. T(n, k) = [z^k] h(n, 1, z) where h(n, v, z) are the modified Lommel polynomials (A369117).

1, 0, 2, -1, 0, 8, 0, -8, 0, 48, 1, 0, -72, 0, 384, 0, 18, 0, -768, 0, 3840, -1, 0, 288, 0, -9600, 0, 46080, 0, -32, 0, 4800, 0, -138240, 0, 645120, 1, 0, -800, 0, 86400, 0, -2257920, 0, 10321920, 0, 50, 0, -19200, 0, 1693440, 0, -41287680, 0, 185794560

Offset: 0

Author

Peter Luschny, Jan 30 2024

Keywords

• sign
• tabl

Examples

			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
		

Links

• David Dickinson, On Lommel and Bessel polynomials, Proc. AMS 5 (1954) 946-956.
• Eric Weisstein's World of Mathematics, Lommel Polynomial.

Crossrefs

Diagonals include: A000165 (main diagonal), A014479, A286725.

Columns include bisections of: A001105, A254371.

Cf. A093985 (row sums), A036243 (abs row sums), A369117.

Programs

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

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).