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

Author

N. J. A. Sloane, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)

Keywords

• sign
• frac
• tabl

Examples

			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.
		

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.

Crossrefs

For denominators see A144618. Cf. A144622.

Programs

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 *)

Extensions

Terms up to u_5 from Chris Kormanyos (ckormanyos(AT)yahoo.com), Jan 18 2009