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.

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

Chris Kormanyos, Rows n=0..121 of triangle, flattened

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

Adjacent sequences: A144614 A144615 A144616 * A144618 A144619 A144620

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