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

2, 16, 198, 2368, 30100, 392544, 5248782, 71501056, 989177508, 13859716000, 196282985756, 2805235913088, 40408113882344, 586055349387200, 8551024115349150, 125431745952519168, 1848653992986172324, 27362153523832614432, 406546456064695351020

Offset: 1

Author

Bradley Klee, Apr 19 2020

Keywords

• nonn
• walk

Comments

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.

References

• B. Klee and É. Angelini, "Social Distancing and A005260", [math-fun] mailing list, Apr. 19, 2020.

Links

• Oskar Schlemmer, Das Triadisches Ballett: Gelbe Marsch, Bavaria Atelier, 1970.
• D. Zeilberger, The Method of Creative Telescoping, Journal of Symbolic Computation, 11.3 (1991), 195-204.

Crossrefs

Cf. A005260.

Programs

Mathematica

RecurrenceTable[{Dot[{(n-1)^2*(n+1)^3*(5*n^2-10*n+4),
-2*n^2*(2*n-1)*(15*n^4-30*n^3+7*n^2+8*n-8),
-4*(n-1)^2*n*(4*n-5)*(4*n-3)*(5*n^2-1)},
a[n-#]&/@Range[0,2]] == 0, a[0] == 0, a[1] == 2},
a, {n, 0, 100}]

PARI

a(n) = 2*sum(k=0, n-1, binomial(n,k)^2*binomial(n,k+1)^2); \\ Michel Marcus, Apr 19 2020

Formula

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.

a(n) ~ 2^(4*n + 3/2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Apr 20 2020