A002462 Coefficients of Legendre polynomials.

1, 1, 9, 50, 1225, 7938, 106722, 736164, 41409225, 295488050, 4266847442, 31102144164, 914057459042, 6760780022500, 100583849722500, 751920156592200, 90324408810638025, 680714748752836050, 10294760089163261250, 78080479568224402500, 2375208188465386324050

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Appears to divide A002894(n+1). - Ralf Stephan, Aug 23 2004

Constant term of the Legendre polynomials of even order when they are expressed in terms of the cosine function (see 22.3.13 from Abramowitz & Stegun) with the denominators factored out. Also, constant term of the Tisserand functions of even order for the planar case with the denominators factored out (see Table 1 from Laskar & Boué's paper). Cf. A002463. - Ruperto Corso, Dec 08 2011

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).

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 776.
• J. Laskar and G. Boué, Explicit expansion of the three-body disturbing function for arbitrary eccentricities and inclinations, arXiv:1008.2947 [astro-ph.IM], 2010; A&A 522, A60 (November 2010).

Programs

Maple

f:=(n,q)->binomial(2*(n-q),(n-q))*binomial(2*q,q)/(4^n): seq(f(2*m,m)*lcm(seq(denom(2*f(2*m,i)), i=0..m-1), denom(f(2*m,m))), m=0..25); # Ruperto Corso, Dec 08 2011

Mathematica

f[n_, q_] := Binomial[2(n-q), n-q] Binomial[2q, q]/4^n;
a[m_] := f[2m, m] LCM @@ Append[Table[Denominator[2f[2m, i]], {i, 0, m-1}], Denominator[f[2m, m]]];
Table[a[m], {m, 0, 25}] (* Jean-François Alcover, Jan 20 2019, after Ruperto Corso *)

Formula

This is binomial(2*n,n)^2/2^(4*n) multiplied by some power of 2, but the exact power of 2 needed is a bit hard to describe precisely. No doubt Prévost or Fletcher et al., where I saw this sequence 40 years ago, will have the answer. - N. J. A. Sloane, Jun 01 2013

Extensions

Sequence extended by Ruperto Corso, Dec 08 2011