A002514 Coefficients in the asymptotic expansions of modified Hankel functions h_1(z) and h_2(z), rounded to nearest integer.

0, 0, 0, 0, 1, 3, 15, 79, 474, 3207, 24087, 198923, 1791902, 17484377, 183707380, 2067904033, 24827519376, 316694549817, 4277112686513, 60971132411393, 914869422343564, 14413525170009350, 237888443951757586, 4104608160094692304

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• The Staff of the Computational Laboratory, Tables of the Modified Hankel Functions of Order One-Third and of Their Derivatives. Annals of the Computation Laboratory of Harvard University, Vol. 2, Harvard Univ. Press, Cambridge, Massachusetts, 1945. see p. XXXV.

Links

• Robert Israel, Table of n, a(n) for n = 1..459
• The Staff of the Computational Laboratory, Tables of the Modified Hankel Functions of Order One-Third and of Their Derivatives. [Annotated scans of two pages]

Programs

Maple

seq(round(simplify(GAMMA(n+5/6)*GAMMA(n+1/6)*3^n/(2^(2*n+1)*Pi*n!))),n=1..50); # Robert Israel, Oct 19 2015

Formula

a(n) = round((Product_{k=1..n} (9 * (2*k-1)^2 - 4)) / (2^(4*n) * 3^n * n!)). - Sean A. Irvine, Oct 18 2015

a(n) = round(Gamma(n+5/6)*Gamma(n+1/6)*3^n/(2^(2*n+1)*Pi*n!)). - Robert Israel, Oct 19 2015

Extensions

More terms from Sean A. Irvine, Oct 18 2015