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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.

%I A171636 #15 Feb 16 2025 08:33:11
%S A171636 2,24,0,1,480,0,16,13440,0,360,0,1,483840,0,10752,0,42,21288960,0,
%T A171636 403200,0,1728,0,1,1107025920,0,18247680,0,79200,0,80,66421555200,0,
%U A171636 968647680,0,4118400,0,5280,0,1,4516665753600,0,59041382400,0,242161920
%N 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.
%C A171636 Lommel polynomials are rational functions and not polynomials.
%H A171636 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LommelPolynomial.html">Lommel Polynomial</a>.
%e A171636 {2},
%e A171636 {24, 0, 1},
%e A171636 {480, 0, 16},
%e A171636 {13440, 0, 360, 0, 1},
%e A171636 {483840, 0, 10752, 0, 42},
%e A171636 {21288960, 0, 403200, 0, 1728, 0, 1},
%e A171636 {1107025920, 0, 18247680, 0, 79200, 0, 80},
%e A171636 {66421555200, 0, 968647680, 0, 4118400, 0, 5280, 0, 1},
%e A171636 {4516665753600, 0, 59041382400, 0, 242161920, 0, 349440, 0, 130},
%e A171636 {343266597273600, 0, 4064999178240, 0, 15968010240, 0, 24460800, 0, 12600, 0, 1}
%p A171636 L := (n, m, z) -> (GAMMA(n + m)/(GAMMA(n)*(z/2)^m))*hypergeom([(1 - m)/2, -m/2],
%p A171636 [n, -m, 1 - n - m], z^2);
%p A171636 for n from 1 to 10 do L(n, n, 1/z): convert(series(%, z, 12), polynom):
%p A171636 lprint(seq(coeff(expand(%), z, n - k), k = 0 .. n - irem(n, 2))): od:
%p A171636 # _Peter Luschny_, Jan 29 2024
%t A171636 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 A171636 Table[CoefficientList[Expand[Lommel[n, n, x]*x^n], x], {n, 1, 10}]
%t A171636 Flatten[%]
%Y A171636 Variant: A369117.
%K A171636 nonn,uned,tabl
%O A171636 1,1
%A A171636 _Roger L. Bagula_, Dec 13 2009