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A337350
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a(n) is the number of lattice paths from (0,0) to (2n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (2k,2k).
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2
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1, 6, 34, 300, 3146, 36244, 443156, 5646040, 74137050, 996217860, 13633173180, 189347631720, 2662142601924, 37815138677960, 541882155414376, 7823955368697776, 113712609033955834, 1662288563798703204, 24424940365489658540, 360537080085493670856
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OFFSET
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0,2
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COMMENTS
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The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.
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LINKS
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Christian Krattenthaler, Lattice path enumeration. In: Handbook of Enumerative Combinatorics. Edited by Miklos Bona. CRC Press, 2015, pages 589-678.
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FORMULA
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G.f.: 2 - 1 / (Sum_{n>=0} binomial(4*n,2*n) * x^n).
Conjecture: a(n) = binomial(4*n,2*n) * (8*n+1) / (8*n^2 + 2*n - 1) for n >= 1.
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PROG
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(PARI) seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(4*k, 2*k)*x^k)))} \\ Andrew Howroyd, Aug 25 2020
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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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