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
Examples
{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}
Links
- Eric Weisstein's World of Mathematics, Lommel Polynomial.
Crossrefs
Variant: A369117.
Programs
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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
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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[%]
Comments