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%I #43 Aug 25 2023 08:39:42
%S 1,0,-6,0,90,0,-1680,0,34650,0,-756756,0,17153136,0,-399072960,0,
%T 9465511770,0,-227873431500,0,5550996791340,0,-136526995463040,0,
%U 3384731762521200,0,-84478098072866400,0,2120572665910728000,0,-53494979785374631680,0
%N Central values of the n-th discrete Chebyshev polynomials of order 2n.
%C In the general case the n-th discrete Chebyshev polynomial of order N is D(N,n;x) = Sum_{i = 0..n} (-1)^i*C(n,i)*C(N-x,n-i)*C(x,i). For N = 2*n , x = n, one gets a(n) = D(2n,n;n) = Sum_{i = 0..n} (-1)^i*C(n,i)^3 that equals (due to Dixon's formula) 0 for odd n and (-1)^m*(3m)!/(m!)^3 for n = 2*m. (Riordan, 1968) So, a(2*m) = (-1)^m*A006480(m).
%D John Riordan, Combinatorial Identities, John Willey&Sons Inc., 1968.
%H Nikita Gogin and Mika Hirvensalo, <a href="http://tucs.fi/publications/attachment.php?fname=TR819.pdf">On the Generating Function of Discrete Chebyshev Polynomials</a>, TUCS Technical Reports 819, Turku Centre for Computer Science, 2007.
%H Nikita Gogin and Mika Hirvensalo, <a href="http://dx.doi.org/10.3233/FI-2012-671">Recurrent Constructions of the MacWilliams and Chebyshev Matrices</a>, Fundamenta Informaticae, vol.116, issue 1-4, January 2012, pages 93-110.
%H H. W. Gould, <a href="http://www.math.wvu.edu/~hgould/">Tables of Combinatorial Identities</a>, Edited by J. Quaintance.
%H Darij Grinberg, <a href="http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf">Introduction to Modern Algebra</a> (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).
%H Y. Sun and X. Wang, <a href="http://www.jstor.org/stable/40378298">A new proof of a curious identity</a>, Math Gazette, Vol. 91, No. 520 (Mar 2007) 105-107.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>
%F a(n) is a coefficient at t^n in (1-t^2)^n*P(0,-(2*n+1);n;(1+t^2)/(1-t^2)), where P(a,b;k;x) is the k-th Jacobi polynomial (Gogin and Hirvensalo, 2007).
%F G.f.: Hypergeometric2F1[1/3,2/3,1,-27*x^2].
%F a(2*m+1) = 0, a(2*m) = (-1)^m*A006480(m).
%F From _Peter Bala_, Aug 04 2016: (Start)
%F a(n) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(2*n - k,n)*binomial(n + k,n) (Sun and Wang).
%F a(n) = Sum_{k = 0..n} (-1)^(n + k)*binomial(n + k, n - k)*binomial(2*k, k)*binomial(2*n - k, n) (Gould, Vol.5, 9.23).
%F a(n) = -1/(n + 1)^3 * A273630(n+1). (End)
%F From _Peter Bala_, Mar 22 2022: (Start)
%F a(n) = - (3*(3*n-2)*(3*n-4)/n^2)*a(n-2).
%F a(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Compare with A002894(n) = binomial(2*n,n)^2 = [x^n] (1 - x)^(2*n) * P(2*n,(1 + x)/(1 - x)). Cf. A103882. (End)
%F From _Peter Bala_, Jul 23 2023: (Start)
%F a(n) = G(x)^(3*n), where the power series G(x) = 1 - x^2 + 2*x^4 - 14*x^6 + 127*x^8 - 1364*x^10 + ... appears to have integer coefficients.
%F exp(Sum_{n >= 1} a(n)*x^n/n) = F(x)^3, where the power series F(x) = 1 - x^2 + 8*x^4 - 101*x^6 + 1569*x^8 - 27445*x^10 + ..., appears to have integer coefficients. See A229452.
%F Row 1 of A364303. (End)
%t Table[Coefficient[Simplify[JacobiP[n,0,-(2*n+1),(1+t^2)/(1-t^2)]*(1-t^2)^n],t,n],{n,0,20}]
%o (Python)
%o from math import factorial
%o def A245086(n): return 0 if n&1 else (-1 if (m:=n>>1)&1 else 1)*factorial(3*m)//factorial(m)**3 # _Chai Wah Wu_, Oct 04 2022
%Y Cf. A000172, A002894, A005260, A006480, A103882, A273630, A364303.
%K sign,easy
%O 0,3
%A _Nikita Gogin_, Jul 11 2014
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