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 A135390 #23 May 09 2025 23:08:50
%S A135390 1,15,310,7455,195426,5416026,156061620,4628393055,140348412490,
%T A135390 4331544836190,135614951248140,4296741195214650,137507314754659500,
%U A135390 4438467396322843500,144329729055650881560,4723733064176346346335
%N A135390 Number of walks from origin to (1,0,0) in a cubic lattice.
%C A135390 a(n) is the number of walks of length 2n+1 on a cubic lattice that start at the origin and end at (1,0,0) using steps (1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1).
%H A135390 Stefan Hollos and Richard Hollos, <a href="http://www.exstrom.com/math/lattice/latpath.html">Lattice Paths and Walks</a>.
%H A135390 Nobu C. Shirai and Naoyuki Sakumichi, <a href="https://arxiv.org/abs/2408.14992">Universal Negative Energetic Elasticity in Polymer Chains: Crossovers among Random, Self-Avoiding, and Neighbor-Avoiding Walks</a>, arXiv:2408.14992 [cond-mat.soft], 2024. See p. 5.
%F A135390 a(n) = binomial(2*n+1,n) * Sum_{k=0..n} binomial(n,k) * binomial(n+1,k) * binomial(2*k,k).
%F A135390 G.f.: (1/(6*z)) * (1/sqrt(1+12*z)*hypergeom([1/8,3/8],[1],64/81*z*(1+sqrt(1-36*z))^2*(2+sqrt(1-36*z))^4/(1+12*z)^4)*hypergeom([1/8,3/8],[1],64/81*z*(1-sqrt(1-36*z))^2*(2-sqrt(1-36*z))^4/(1+12*z)^4) - 1). - _Sergey Perepechko_, Jan 31 2011
%F A135390 a(n) = A002896(n+1)/6.
%t A135390 f[n_] := Binomial[2 n + 1, n]*Sum[ Binomial[n, k]*Binomial[n + 1, k]*Binomial[2 k, k], {k, 0, n}]; Array[f, 16, 0] (* _Robert G. Wilson v_ *)
%o A135390 (Maxima) a(n) = binomial(2*n+1,n) * sum( binomial(n,k) * binomial(n+1,k) * binomial(2*k,k), k, 0, n );
%Y A135390 Cf. A002896.
%K A135390 easy,nonn
%O A135390 0,2
%A A135390 Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007