A369117 -id:A369117 - cp's OEIS frontend

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

Peter Luschny, Jan 30 2024

			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

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

Diagonals include: A000165 (main diagonal), A014479, A286725.
Columns include bisections of: A001105, A254371.
Cf. A093985 (row sums), A036243 (abs row sums), A369117.

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);

Table[CoefficientList[Expand[ResourceFunction["LommelR"][n, 1, 1/z]], z], {n, 0, 8}] // MatrixForm

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).

A171636 Table read by rows. Coefficients of Lommel polynomials L(n, m, z) = (Gamma(n + m) / (Gamma(n) * (z/2)^m)) * hypergeom([(1 - m)/2, -m/2], [n, -m, 1 - n - m], z^2) for n = m and descending powers.

Original entry on oeis.org

2, 24, 0, 1, 480, 0, 16, 13440, 0, 360, 0, 1, 483840, 0, 10752, 0, 42, 21288960, 0, 403200, 0, 1728, 0, 1, 1107025920, 0, 18247680, 0, 79200, 0, 80, 66421555200, 0, 968647680, 0, 4118400, 0, 5280, 0, 1, 4516665753600, 0, 59041382400, 0, 242161920

Offset: 1

Roger L. Bagula, Dec 13 2009

Lommel polynomials are rational functions and not polynomials.

			{2},
{24, 0, 1},
{480, 0, 16},
{13440, 0, 360, 0, 1},
{483840, 0, 10752, 0, 42},
{21288960, 0, 403200, 0, 1728, 0, 1},
{1107025920, 0, 18247680, 0, 79200, 0, 80},
{66421555200, 0, 968647680, 0, 4118400, 0, 5280, 0, 1},
{4516665753600, 0, 59041382400, 0, 242161920, 0, 349440, 0, 130},
{343266597273600, 0, 4064999178240, 0, 15968010240, 0, 24460800, 0, 12600, 0, 1}

Eric Weisstein's World of Mathematics, Lommel Polynomial.

Variant: A369117.

L := (n, m, z) -> (GAMMA(n + m)/(GAMMA(n)*(z/2)^m))*hypergeom([(1 - m)/2, -m/2],
[n, -m, 1 - n - m], z^2);
for n from 1 to 10 do L(n, n, 1/z): convert(series(%, z, 12), polynom):
lprint(seq(coeff(expand(%), z, n - k), k = 0 .. n - irem(n, 2))): od:
# Peter Luschny, Jan 29 2024

Lommel[m_, n_, z_] := (Gamma[n + m]/(Gamma[n] ((z/ 2))^m)) HypergeometricPFQ[{((1 - m))/2, (- m)/2}, {n, (-m), 1 - n - m}, z^2]
Table[CoefficientList[Expand[Lommel[n, n, x]*x^n], x], {n, 1, 10}]
Flatten[%]

cp's OEIS Frontend

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).

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