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A144617 Triangle read by rows: numerators of coefficients of the Debye-type polynomial u_n used for asymptotic Airy-type expansions of Bessel functions of arbitrary large order.

%I A144617 #15 Mar 31 2025 11:57:48
%S A144617 1,3,-5,81,-462,385,30375,-369603,765765,-425425,4465125,-94121676,
%T A144617 349922430,-446185740,185910725,1519035525,-49286948607,284499769554,
%U A144617 -614135872350,566098157625,-188699385875
%N A144617 Triangle read by rows: numerators of coefficients of the Debye-type polynomial u_n used for asymptotic Airy-type expansions of Bessel functions of arbitrary large order.
%H A144617 Chris Kormanyos, <a href="/A144617/b144617.txt">Rows n=0..121 of triangle, flattened</a>
%H A144617 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972. See Section 9.3.9.
%H A144617 M. Abramowitz and I. A. Stegun, eds., <a href="https://personal.math.ubc.ca/~cbm/aands/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. See Section 9.3.9.
%e A144617 The polynomials u_0, u_1, u_2 and u_3 are:
%e A144617 1;
%e A144617 (3*t - 5*t^3)/24;
%e A144617 (81*t^2 - 462*t^4 + 385*t^6)/1152;
%e A144617 (30375*t^3 - 369603*t^5 + 765765*t^7 - 425425*t^9)/414720.
%t A144617 uktop = {1, 3, -5}; ukbot = {1, 24}; u = ((3 t) - (5 (t^3)))/24; Do[uk = (((1/2) (t^2) (1 - (t^2))) D[u, t]) + ((1/8) Integrate[((1 - (5 (t^2))) u), {t, 0, t}]); u = Simplify[uk]; Do[uktop = Append[uktop, Coefficient[Expand[Numerator[u]], t^n]], {n, k, 3 k, 2}]; ukbot = Append[ukbot, Denominator[u]]; Print[k], {k, 2, 8}]; (* Chris Kormanyos (ckormanyos(AT)yahoo.com), Jan 18 2009 *)
%Y A144617 For denominators see A144618. Cf. A144622.
%K A144617 sign,frac,tabl
%O A144617 0,2
%A A144617 _N. J. A. Sloane_, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)
%E A144617 Terms up to u_5 from Chris Kormanyos (ckormanyos(AT)yahoo.com), Jan 18 2009