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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.
3
1, 3, -5, 81, -462, 385, 30375, -369603, 765765, -425425, 4465125, -94121676, 349922430, -446185740, 185910725, 1519035525, -49286948607, 284499769554, -614135872350, 566098157625, -188699385875
OFFSET
0,2
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972. See Section 9.3.9.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. See Section 9.3.9.
EXAMPLE
The polynomials u_0, u_1, u_2 and u_3 are:
1;
(3*t - 5*t^3)/24;
(81*t^2 - 462*t^4 + 385*t^6)/1152;
(30375*t^3 - 369603*t^5 + 765765*t^7 - 425425*t^9)/414720.
MATHEMATICA
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 *)
CROSSREFS
For denominators see A144618. Cf. A144622.
Sequence in context: A062214 A323490 A368015 * A301492 A107655 A182234
KEYWORD
sign,frac,tabl
AUTHOR
N. J. A. Sloane, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)
EXTENSIONS
Terms up to u_5 from Chris Kormanyos (ckormanyos(AT)yahoo.com), Jan 18 2009
STATUS
approved