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A269877
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A double binomial sum involving absolute values.
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2
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0, 8, 121728, 77214720, 12676235264, 1090372239360, 64922717257728, 3052335748087808, 121762580539637760, 4304417014325182464, 138706918527488491520, 4154140250223566389248, 117243264067548833906688, 3150495258536853477785600, 81236017376284183797694464
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OFFSET
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0,2
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COMMENTS
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A fast algorithm follows from Theorem 5 of Brent et al. article.
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LINKS
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FORMULA
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G.f.: 8*x*(1 + 15104*x + 7953024*x^2 + 585181184*x^3 + 8538456064*x^4 + 19750453248*x^5)/(1-16*x)^7.
a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*(k^2 - l^2)^6).
a(n) = 2^(4*n-3)*n*(2*n-1)*(900*n^4-4500*n^3+8895*n^2-8055*n+2764).
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MATHEMATICA
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Table[2^(4 n - 3) n (2 n - 1) (900 n^4 - 4500 n^3 + 8895 n^2 - 8055 n + 2764), {n, 0, 15}]
LinearRecurrence[{112, -5376, 143360, -2293760, 22020096, -117440512, 268435456}, {0, 8, 121728, 77214720, 12676235264, 1090372239360, 64922717257728}, 20] (* Harvey P. Dale, Oct 28 2023 *)
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PROG
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(Magma) [2^(4*n-3)*n*(2*n-1)*(900*n^4-4500*n^3+8895*n^2-8055*n+2764): n in [0..20]];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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