This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337350 #48 Jul 16 2025 14:57:58
%S A337350 1,6,34,300,3146,36244,443156,5646040,74137050,996217860,13633173180,
%T A337350 189347631720,2662142601924,37815138677960,541882155414376,
%U A337350 7823955368697776,113712609033955834,1662288563798703204,24424940365489658540,360537080085493670856
%N A337350 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 A337350 The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.
%H A337350 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.
%H A337350 R. J. Mathar, <a href="https://vixra.org/abs/2502.0097">The Eggenberger-Polya urn process: Probabilities of revisited ball ratios</a>, vixra:2502.0097 (2025) Table 4
%H A337350 Quy Nhan, <a href="https://math.stackexchange.com/q/5077486">A finite sum of binomial coefficients</a>, MathStackExchange, 2025-06-21.
%F A337350 G.f.: 2 - 1 / (Sum_{n>=0} binomial(4*n,2*n) * x^n).
%F A337350 a(n) = binomial(4*n,2*n) * (8*n+1) / (8*n^2 + 2*n - 1) for n >= 1. For proof, see the Quy Nhan link.
%F A337350 D-finite with recurrence n*(2*n+1)*(8*n-7)*a(n) -2*(4*n-5)*(4*n-3)*(8*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 26 2025
%F A337350 From _Lucas A. Brown_, Jul 13 2025: (Start)
%F A337350 G.f.: 2 - sqrt(2-32*x) / sqrt(1+sqrt(1-16*x)).
%F A337350 a(n) = A000108(2*n) + 4 * A000108(2*n-1). (End)
%o A337350 (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 A337350 Cf. A000108, A001448, A337291, A337292, A337351, A337352.
%K A337350 nonn,easy
%O A337350 0,2
%A A337350 _Lucas A. Brown_, Aug 24 2020