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A002462
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Coefficients of Legendre polynomials.
(Formerly M4633 N1979)
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1
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1, 1, 9, 50, 1225, 7938, 106722, 736164, 41409225, 295488050, 4266847442, 31102144164, 914057459042, 6760780022500, 100583849722500, 751920156592200, 90324408810638025, 680714748752836050, 10294760089163261250, 78080479568224402500, 2375208188465386324050
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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