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a(n) is the number of lattice paths from (0,0) to (3n,3n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (3k,3k).
2

%I #15 Sep 20 2020 00:45:31

%S 1,20,524,19660,854380,40304080,2004409236,103440770760,5486614131756,

%T 297239307415792,16376472734974384,914734188877259884,

%U 51680064605716043636,2948046519564292501232,169560941932509940657016,9822377923336683964009296,572554753384166308597716396

%N a(n) is the number of lattice paths from (0,0) to (3n,3n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (3k,3k).

%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(6*n,3*n) * x^n).

%o (PARI) seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(6*k,3*k)*x^k)))} \\ _Andrew Howroyd_, Aug 25 2020

%Y Cf. A337291, A337292, A337350, A337351.

%K nonn,easy

%O 0,2

%A _Lucas A. Brown_, Aug 24 2020