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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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%I #14 Jan 29 2024 19:19:37

%S 2,24,0,1,480,0,16,13440,0,360,0,1,483840,0,10752,0,42,21288960,0,

%T 403200,0,1728,0,1,1107025920,0,18247680,0,79200,0,80,66421555200,0,

%U 968647680,0,4118400,0,5280,0,1,4516665753600,0,59041382400,0,242161920

%N 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.

%C Lommel polynomials are rational functions and not polynomials.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LommelPolynomial.html">Lommel Polynomial</a>.

%e {2},

%e {24, 0, 1},

%e {480, 0, 16},

%e {13440, 0, 360, 0, 1},

%e {483840, 0, 10752, 0, 42},

%e {21288960, 0, 403200, 0, 1728, 0, 1},

%e {1107025920, 0, 18247680, 0, 79200, 0, 80},

%e {66421555200, 0, 968647680, 0, 4118400, 0, 5280, 0, 1},

%e {4516665753600, 0, 59041382400, 0, 242161920, 0, 349440, 0, 130},

%e {343266597273600, 0, 4064999178240, 0, 15968010240, 0, 24460800, 0, 12600, 0, 1}

%p L := (n, m, z) -> (GAMMA(n + m)/(GAMMA(n)*(z/2)^m))*hypergeom([(1 - m)/2, -m/2],

%p [n, -m, 1 - n - m], z^2);

%p for n from 1 to 10 do L(n, n, 1/z): convert(series(%, z, 12), polynom):

%p lprint(seq(coeff(expand(%), z, n - k), k = 0 .. n - irem(n, 2))): od:

%p # _Peter Luschny_, Jan 29 2024

%t 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]

%t Table[CoefficientList[Expand[Lommel[n, n, x]*x^n], x], {n, 1, 10}]

%t Flatten[%]

%Y Variant: A369117.

%K nonn,uned,tabl

%O 1,1

%A _Roger L. Bagula_, Dec 13 2009