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%I #22 Sep 20 2020 00:45:07
%S 1,6,34,300,3146,36244,443156,5646040,74137050,996217860,13633173180,
%T 189347631720,2662142601924,37815138677960,541882155414376,
%U 7823955368697776,113712609033955834,1662288563798703204,24424940365489658540,360537080085493670856
%N 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).
%C The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.
%H Christian Krattenthaler, <a href="https://www.mat.univie.ac.at/~kratt/artikel/encylatt.pdf">Lattice path enumeration</a>. In: Handbook of Enumerative Combinatorics. Edited by Miklos Bona. CRC Press, 2015, pages 589-678.
%F G.f.: 2 - 1 / (Sum_{n>=0} binomial(4*n,2*n) * x^n).
%F Conjecture: a(n) = binomial(4*n,2*n) * (8*n+1) / (8*n^2 + 2*n - 1) for n >= 1.
%o (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
%Y Cf. A337291, A337292, A337351, A337352.
%K nonn,easy
%O 0,2
%A _Lucas A. Brown_, Aug 24 2020
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