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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
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: A308646 A338510 A221246 * A210061 A034814 A034816
KEYWORD
nonn
AUTHOR
EXTENSIONS
Sequence extended by Ruperto Corso, Dec 08 2011
STATUS
approved

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Last modified March 29 09:32 EDT 2024. Contains 371268 sequences. (Running on oeis4.)