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A002462 Coefficients of Legendre polynomials.
(Formerly M4633 N1979)
1
1, 1, 9, 50, 1225, 7938, 106722, 736164, 41409225, 295488050, 4266847442, 31102144164, 914057459042, 6760780022500, 100583849722500, 751920156592200, 90324408810638025, 680714748752836050, 10294760089163261250, 78080479568224402500, 2375208188465386324050 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Table of n, a(n) for n=0..20.

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).

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

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 *)

CROSSREFS

Sequence in context: A115366 A188210 A221246 * A210061 A034814 A034816

Adjacent sequences:  A002459 A002460 A002461 * A002463 A002464 A002465

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Sequence extended by Ruperto Corso, Dec 08 2011

STATUS

approved

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Last modified March 19 15:02 EDT 2019. Contains 321330 sequences. (Running on oeis4.)