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.

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

1,1

Comments

Lommel polynomials are rational functions and not polynomials.

Links

Table of n, a(n) for n=1..45.

Eric Weisstein's World of Mathematics, Lommel Polynomial.

Example

{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}

Maple

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

Mathematica

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[%]

Crossrefs

Variant: A369117.

Sequence in context: A118812 A228241 A054909 * A375304 A350257 A270562

Adjacent sequences: A171633 A171634 A171635 * A171637 A171638 A171639

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, Dec 13 2009

Status

approved