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A334237 a(n) = 2*Sum_{k=0..n-1} binomial(n,k)^2*binomial(n,k+1)^2.

%I A334237 #30 Jul 15 2022 02:30:53
%S A334237 2,16,198,2368,30100,392544,5248782,71501056,989177508,13859716000,
%T A334237 196282985756,2805235913088,40408113882344,586055349387200,
%U A334237 8551024115349150,125431745952519168,1848653992986172324,27362153523832614432,406546456064695351020
%N A334237 a(n) = 2*Sum_{k=0..n-1} binomial(n,k)^2*binomial(n,k+1)^2.
%C A334237 a(n) is also the number of simultaneous walks between two walkers on an n X n grid, subject to a "social distancing" constraint. The rules are the same as in A005260, but the counting criterion is changed so that the walkers cannot meet. Instead, they must be separated by closest-approach distance of sqrt(2) after n steps. Each term a(n) is a hypergeometric single sum, so Zeilberger's algorithm applies, and a(n) must also satisfy a p-recurrence.
%D A334237 B. Klee and É. Angelini, "Social Distancing and A005260", [math-fun] mailing list, Apr. 19, 2020.
%H A334237 Oskar Schlemmer, <a href="https://www.youtube.com/watch?v=mHQmnumnNgo&amp;feature=youtu.be&amp;t=258">Das Triadisches Ballett: Gelbe Marsch</a>, Bavaria Atelier, 1970.
%H A334237 D. Zeilberger, <a href="https://doi.org/10.1016/S0747-7171(08)80044-2">The Method of Creative Telescoping</a>, Journal of Symbolic Computation, 11.3 (1991), 195-204.
%F A334237 D-finite with recurrence (n-1)^2*(n+1)^3*(5*n^2-10*n+4)*a(n) - 2*n^2*(2*n-1)*(15*n^4-30*n^3+7*n^2+8*n-8)*a(n-1) - 4*(n-1)^2*n*(4*n-5)*(4*n-3)*(5*n^2-1)*a(n-2) = 0.
%F A334237 a(n) ~ 2^(4*n + 3/2) / (Pi*n)^(3/2). - _Vaclav Kotesovec_, Apr 20 2020
%t A334237 RecurrenceTable[{Dot[{(n-1)^2*(n+1)^3*(5*n^2-10*n+4),
%t A334237 -2*n^2*(2*n-1)*(15*n^4-30*n^3+7*n^2+8*n-8),
%t A334237 -4*(n-1)^2*n*(4*n-5)*(4*n-3)*(5*n^2-1)},
%t A334237 a[n-#]&/@Range[0,2]] == 0, a[0] == 0, a[1] == 2},
%t A334237 a, {n, 0, 100}]
%o A334237 (PARI) a(n) = 2*sum(k=0, n-1, binomial(n,k)^2*binomial(n,k+1)^2); \\ _Michel Marcus_, Apr 19 2020
%Y A334237 Cf. A005260.
%K A334237 nonn,walk
%O A334237 1,1
%A A334237 _Bradley Klee_, Apr 19 2020