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A143010
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Crystal ball sequence for the A4 x A4 lattice.
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4
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1, 41, 661, 5741, 33001, 142001, 494341, 1465661, 3833941, 9073501, 19789001, 40328641, 77620661, 142282141, 250054001, 423621001, 694880441, 1107728161, 1721435341, 2614694501, 3890418001, 5681377241, 8156775661, 11529853541
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OFFSET
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0,2
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COMMENTS
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The A_4 lattice consists of all vectors v = (a,b,c,d,e) in Z^5 such that a+b+c+d+e = 0. The lattice is equipped with the norm ||v|| = 1/2*(|a| + |b| + |c| + |d| + |e|). Pairs of lattice points (v,w) in the product lattice A_4 x A_4 have norm ||(v,w)|| = ||v|| + ||w||. Then the k-th term in the crystal ball sequence for the A_4 x A_4 lattice gives the number of such pairs (v,w) for which ||(v,w)|| is less than or equal to k.
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LINKS
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FORMULA
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a(n) = (35*n^8 +140*n^7 +630*n^6 +1400*n^5 +2595*n^4 +3020*n^3 +2500*n^2 +1200*n +288)/288 = 5*n*(n + 1)*(n^2 + n + 2)*(7*n^4 + 14*n^3 + 77*n^2 + 70*n + 120)/288 + 1.
O.g.f. : 1/(1-x)*[Legendre_P(4,(1+x)/(1-x))]^2.
Apery's constant zeta(3) = (1+1/2^3+1/3^3+1/4^3) + Sum {n = 1..inf} 1/(n^3*a(n-1)*a(n)).
G.f.: (1+16*x+36*x^2+16*x^3+x^4)^2/(1-x)^9. [Colin Barker, Mar 16 2012]
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8. - Vincenzo Librandi, Dec 16 2015
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MAPLE
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p := n -> (35*n^8 +140*n^7 +630*n^6 +1400*n^5 +2595*n^4 +3020*n^3 +2500*n^2 +1200*n +288)/288: seq(p(n), n = 0..24);
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MATHEMATICA
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LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 41, 661, 5741, 33001, 142001, 494341, 1465661, 3833941}, 25] (* Vincenzo Librandi, Dec 16 2015 *)
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PROG
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(Python)
A143010_list, m = [], [4900, -14700, 17500, -10500, 3340, -540, 40, 0, 1]
for _ in range(10**2):
for i in range(8):
(Magma) [5*n*(n+1)*(n^2+n+2)*(7*n^4+14*n^3+77*n^2+70*n+120)/288+1: n in [0..30]]; // Vincenzo Librandi, Dec 16 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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