A002463 Coefficients of Legendre polynomials.

1, 3, 30, 175, 4410, 29106, 396396, 2760615, 156434850, 1122854590, 16291599324, 119224885962, 3515605611700, 26077294372500, 388924218927000, 2913690606794775, 350671234206006450, 2647224022927695750, 40095381399899017500, 304513870316075169750

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Apparently, a(n) divides A000894(n). - Ralf Stephan, Aug 05 2004

Coefficients of cos(x) term of the Tisserand functions of odd order for the planar case with the denominators factored out (see Table 1 from Laskar & Boué's paper) (cf A002462). - Michel Marcus, May 29 2013

Also cos(x) term of the Legendre polynomials of odd order when they are expressed in terms of the cosine function (see 22.3.13 from Abramowitz & Stegun) with the denominators factored out. - Michel Marcus, May 29 2013

References

• A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 362.
• G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
• 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).

Programs

PARI

lista(nn) = {forstep (n=1, nn, 2, lcmc = 1; for (m=0, n\2, lcmc = lcm(lcmc, denominator(binomial(2*n-2*m, n-m) * binomial(2*m, m)/4^n));); m = n\2; print1(lcmc*binomial(2*n-2*m, n-m) * binomial(2*m, m)/4^n, ", "););} \\ Michel Marcus, May 29 2013

Python

from sympy import binomial as C, lcm
def a_list(nn):
    l = []
    for n in range(1, nn + 1, 2):
        lcmc = 1
        for m in range(n//2 + 1):
            lcmc = lcm(lcmc, (C(2*n - 2*m, n - m)*C(2*m, m)/4**n).denominator)
        m = n//2
        l.append(lcmc*C(2*n - 2*m, n - m)*C(2*m, m)//4**n)
    return l # Indranil Ghosh, Jul 02 2017, after PARI code by Michel Marcus

Extensions

More terms from Michel Marcus, May 29 2013