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%I M4633 N1979 #44 Jan 20 2019 07:46:53
%S 1,1,9,50,1225,7938,106722,736164,41409225,295488050,4266847442,
%T 31102144164,914057459042,6760780022500,100583849722500,
%U 751920156592200,90324408810638025,680714748752836050,10294760089163261250,78080479568224402500,2375208188465386324050
%N Coefficients of Legendre polynomials.
%C Appears to divide A002894(n+1). - _Ralf Stephan_, Aug 23 2004
%C 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
%D 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.
%D G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H M. Abramowitz and I. A. Stegun, eds.,<a href="http://people.math.sfu.ca/~cbm/aands/page_776.htm"> Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 776.
%H J. Laskar and G. Boué, <a href="http://arxiv.org/abs/1008.2947">Explicit expansion of the three-body disturbing function for arbitrary eccentricities and inclinations</a>, arXiv:1008.2947 [astro-ph.IM], 2010; A&A 522, A60 (November 2010).
%F 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
%p 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
%t f[n_, q_] := Binomial[2(n-q), n-q] Binomial[2q, q]/4^n;
%t a[m_] := f[2m, m] LCM @@ Append[Table[Denominator[2f[2m, i]], {i, 0, m-1}], Denominator[f[2m, m]]];
%t Table[a[m], {m, 0, 25}] (* _Jean-François Alcover_, Jan 20 2019, after _Ruperto Corso_ *)
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E Sequence extended by _Ruperto Corso_, Dec 08 2011
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