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 A143011 Crystal ball sequence for the A_5 x A_5 lattice. 5
 1, 61, 1441, 17861, 142001, 819005, 3713305, 13980205, 45432805, 131091505, 342981013, 826861993, 1859914733, 3942293993, 7937011013, 15276834025, 28261896025, 50477521525, 87368496025, 147013666525, 241153442041, 386532523301, 606631094081, 933869816501 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The A_5 lattice consists of all vectors v = (x_1,...,x_6) in Z^6 such that sum {i = 1..6} x_i = 0. The lattice is equipped with the norm ||v|| = 1/2*(sum {i = 1..6} |x_i|). Pairs of lattice points (v,w) in the product lattice A_5 x A_5 have norm ||(v,w)|| = ||v|| + ||w||. Then the k-th term in the crystal ball sequence for the A_5 x A_5 lattice gives the number of such pairs (v,w) for which ||(v,w)|| is less than or equal to k. LINKS T. D. Noe, Table of n, a(n) for n = 0..1000 R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142. FORMULA Row 5 of A143007. a(n) := (126*n^10 +630*n^9 +4095*n^8 +12600*n^7 +36148*n^6 +66990*n^5 +100555*n^4 +102900*n^3 +75076*n^2 +32880*n +7200)/7200. O.g.f. : 1/(1-x)*[Legendre_P(5,(1+x)/(1-x))]^2. Apery's constant zeta(3) = (1+1/2^3+1/3^3+1/4^3+1/5^3) + sum {n = 1..inf} 1/(n^3*a(n-1)*a(n)). G.f.: (1+x)^2*(1+24*x+76*x^2+24*x^3+x^4)^2/(1-x)^11. [Colin Barker, Apr 16 2012] MAPLE p := n -> (126*n^10 +630*n^9 +4095*n^8 +12600*n^7 +36148*n^6 +66990*n^5 +100555*n^4 +102900*n^3 +75076*n^2 +32880*n +7200)/7200: seq(p(n), n = 0..24); CROSSREFS Cf. A143007, A143008, A143009, A143010. Sequence in context: A262017 A060061 A000507 * A211213 A229667 A242382 Adjacent sequences:  A143008 A143009 A143010 * A143012 A143013 A143014 KEYWORD easy,nonn AUTHOR Peter Bala, Jul 22 2008 STATUS approved

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Last modified December 1 16:44 EST 2021. Contains 349430 sequences. (Running on oeis4.)