A369585 - OEIS

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

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..54.

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

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

Columns include bisections of: A001105, A254371.

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

Sequence in context: A021896 A188835 A217735 * A075615 A195284 A351263

Adjacent sequences: A369582 A369583 A369584 * A369586 A369587 A369588

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Jan 30 2024

STATUS

approved