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A171636
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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.
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0
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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
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OFFSET
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1,1
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COMMENTS
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Lommel polynomials are rational functions and not polynomials.
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LINKS
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EXAMPLE
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{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}
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MAPLE
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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:
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MATHEMATICA
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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[%]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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