A369117 Table read by rows. T(n, k) = [z^k] LommelR(n, n, 1/z) where LommelR are the Lommel polynomials.
1, 0, 2, -1, 0, 24, 0, -16, 0, 480, 1, 0, -360, 0, 13440, 0, 42, 0, -10752, 0, 483840, -1, 0, 1728, 0, -403200, 0, 21288960, 0, -80, 0, 79200, 0, -18247680, 0, 1107025920, 1, 0, -5280, 0, 4118400, 0, -968647680, 0, 66421555200, 0, 130, 0, -349440, 0, 242161920, 0, -59041382400, 0, 4516665753600
Offset: 0
Examples
List of coefficients starts: [0] 1; [1] 0, 2; [2] -1, 0, 24; [3] 0, -16, 0, 480; [4] 1, 0, -360, 0, 13440; [5] 0, 42, 0, -10752, 0, 483840; [6] -1, 0, 1728, 0, -403200, 0, 21288960; [7] 0, -80, 0, 79200, 0, -18247680, 0, 1107025920; [8] 1, 0, -5280, 0, 4118400, 0, -968647680, 0, 66421555200;
References
- Eugen von Lommel, Zur Theorie der Bessel'schen Functionen, Math. Ann. 4, 103-116 (1871).
Links
- David Dickinson, On Lommel and Bessel polynomials, AMS Proceedings 1954.
- Eric Weisstein's World of Mathematics, Lommel Polynomial.
Programs
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Maple
Lommel_h := proc(n) local L, k; if n = 0 then return 1 fi; h := (n, m, z) -> (GAMMA(n + m)/(GAMMA(n)*(z/2)^m))*hypergeom([(1 - m)/2, -m/2], [n, -m, 1 - n - m], z^2); convert(series(h(n, n, 1/z), z, n + 1), polynom): seq((-1)^binomial(n-k, 2)*coeff(expand(%), z, k), k = 0..n) end: for n from 0 to 9 do Lommel_h(n) od; # Alternative, by recursion: h := proc(n, v, x) option remember; if n = -1 then 0 elif n = 0 then 1 else 2*(v + n - 1)*z*h(n - 1, v, z) - h(n - 2, v, z) fi end: for n from 0 to 6 do seq(coeff(h(n, n, z), z, k), k = 0..n) end;
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Mathematica
Table[CoefficientList[Expand[ResourceFunction["LommelR"][n, n, 1/z]], z], {n, 0, 9}] // Flatten
Formula
T(n, k) = (-1)^binomial(n - k, 2) * (2*z)^n * [z^k] ((Gamma(2*n)/Gamma(n)) * hypergeom([(1-n)/2, -n/2], [n, -n, 1 - 2*n], z^(-2))) for n > 0 and T(0, 0) = 1.
T(n, k) = [z^k] h(n, n, z) where h(n, v, x) are the modified Lommel polynomials defined by the recurrence h(n, v, x) = 2*(v + n - 1)*z*h(n - 1, v, z) - h(n - 2, v, z) with base values h(-1, v, x) = 0 and h(0, v, x) = 1.
Comments