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A245086 Central values of the n-th discrete Chebyshev polynomials of order 2n. 4
1, 0, -6, 0, 90, 0, -1680, 0, 34650, 0, -756756, 0, 17153136, 0, -399072960, 0, 9465511770, 0, -227873431500, 0, 5550996791340, 0, -136526995463040, 0, 3384731762521200, 0, -84478098072866400, 0, 2120572665910728000, 0, -53494979785374631680, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

In general case the n-th discrete Chebyshev polynomial of order N is D(N,n;x)=Sum[(-1)^i*C(n,i)*C(N-x,n-i)*C(x,i), {i,0,n}] . For N=2n , x=n one gets a(n)=D(2n,n;n)=Sum[(-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=2m. (Riordan, 1968) So, a(2m)=(-1)^m*A006480(m).

REFERENCES

John Riordan, Combinatorial Identities, John Willey&Sons Inc., 1968.

LINKS

Table of n, a(n) for n=0..31.

Nikita Gogin, Mika Hirvensalo, On the Generating Function of Discrete Chebyshev Polynomials, TUCS Technical Reports 819, Turku Centre for Computer Science, 2007.

Nikita Gogin, Mika Hirvensalo, Recurrent Constructions of the MacWilliams and Chebyshev Matrices, Fundamenta Informaticae, vol.116, issue 1-4, January 2012, pages 93-110.

H. W. Gould, Tables of Combinatorial Identities, Edited by J. Quaintance.

Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).

Y. Sun and X. Wang, A new proof of a curious identity, Math Gazette, Vol. 91, No. 520 (Mar 2007) 105-107.

Wikipedia, Dixon's identity

FORMULA

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).

G.f.: Hypergeometric2F1[1/3,2/3,1,-27*x^2].

a(2m+1) = 0, a(2m)=(-1)^m*A006480(m).

From Peter Bala, Aug 04 2016: (Start)

a(n) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(2*n - k,n)*binomial(n + k,n) (Sun and Wang).

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).

a(n) = -1/(n + 1)^3 * A273630(n+1). (End)

MATHEMATICA

Table[Coefficient[Simplify[JacobiP[n, 0, -(2*n+1), (1+t^2)/(1-t^2)]*(1-t^2)^n], t, n], {n, 0, 20}]

CROSSREFS

Cf. A006480, A273630.

Sequence in context: A199044 A156488 A057399 * A145223 A219948 A072129

Adjacent sequences:  A245083 A245084 A245085 * A245087 A245088 A245089

KEYWORD

sign,easy

AUTHOR

Nikita Gogin, Jul 11 2014

STATUS

approved

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)