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A098274
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Sum_{k = 0..n} C(n, k)^2*C(n+k, n)*C(n+2*k, n).
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1
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1, 7, 163, 5191, 191251, 7665757, 324610399, 14287393351, 647133545107, 29966682138757, 1412267381766913, 67516263948005341, 3266295503353540399, 159606073670867165713, 7866072852462175900663
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OFFSET
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0,2
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COMMENTS
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Coefficients in a simultaneous approximation to zeta(2) and zeta(3).
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LINKS
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Table of n, a(n) for n=0..14.
W. Zudilin, Approximations to -, di- and tri-logarithms, arXiv:math/0409023 [math.CA], 2004-2005.
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FORMULA
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a(n) = (-1)^n * Sum_{k = 0..n} (-1)^k*C(n, k)*C(n+k, k)^3.
From Peter Bala, Jan 18 2020: (Start)
a(n) = Sum_{0 <= j, k <= n} (-1)^(j+k)*C(n, k)*C(n, j)*C(n+k, k)*C(n+k+j, k+j)^2.
a(n) = Sum_{0 <= j, k <= n} (-1)^(n+k)*C(n, k)*C(n, j)^2*C(n+k, k)*C(n+k+j, k+j).
(End)
a(n) = hypergeom([1/2+n/2, 1+n/2, -n, -n, 1+n], [1/2, 1, 1, 1], 1). - Peter Luschny, Jan 19 2020
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MAPLE
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a := n -> hypergeom([1/2 + n/2, 1 + n/2, -n, -n, 1 + n], [1/2, 1, 1, 1], 1):
seq(simplify(a(n)), n=0..14); # Peter Luschny, Jan 19 2020
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MATHEMATICA
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Table[Sum[Binomial[n, k]^2 Binomial[n+k, n]Binomial[n+2k, n], {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Aug 17 2019 *)
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PROG
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(MAGMA) [&+[Binomial(n, k)^2 * Binomial(n+k, n) * Binomial(n+2*k, n): k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Jan 19 2020
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CROSSREFS
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Cf. A005258, A005259, A098275.
Sequence in context: A201179 A218998 A220921 * A027549 A212856 A351610
Adjacent sequences: A098271 A098272 A098273 * A098275 A098276 A098277
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KEYWORD
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nonn,easy
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AUTHOR
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Ralf Stephan, Sep 03 2004
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STATUS
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approved
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