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 A337900 #16 Dec 21 2024 22:29:27 %S A337900 1,16,225,3136,44100,627264,9018009,130873600,1914762564,28210561600, %T A337900 418151049316,6230734868736,93271169290000,1401915345465600, %U A337900 21147754404155625,320042195924198400,4857445984927644900,73916947787011560000,1127482124965160372100 %N A337900 The number of walks of length 2n on the square lattice that start from the origin (0,0) and end at the vertex (2,0). %H A337900 R. J. Mathar, <a href="/A337869/a337869.pdf">Random Walk on the Square Lattice: Return to (0,0) with or without passing (1,0)</a> (Sep 2020) %F A337900 a(n) = [A001791(n)]^2. %F A337900 G.f.: x*4F3(3/2, 3/2, 2, 2; 1, 3, 3; 16*x). %F A337900 D-finite with recurrence (n-1)^2*(n+1)^2*a(n) - 4*n^2*(2*n-1)^2*a(n-1) = 0. %F A337900 a(n) = (2n)!*[x^(2n)] BesselI(0, 2x)*BesselI(2, 2x). - _Peter Luschny_, Dec 05 2024 %e A337900 a(2) = 16 counts the walks RRRL, RRLR, RLRR, LRRR, RRUD, RRDU, RDRU, RURD, RUDR, RDUR, URRD, DRRU, URDR, DRUR, UDRR, DURR of length 4. %p A337900 egf := BesselI(0, 2*x)*BesselI(2, 2*x): ser := series(egf, x, 40): %p A337900 seq((2*n)!*coeff(ser, x, 2*n), n = 1..19); # _Peter Luschny_, Dec 05 2024 %Y A337900 Cf. A002894 (at (0,0)), A060150 (at (1,0)), A135389 (at (1,1)), A337901 (at (3,0)), A337902 (at (2,1)). %Y A337900 Cf. A001791. %K A337900 nonn,easy,walk %O A337900 1,2 %A A337900 _R. J. Mathar_, Sep 29 2020