A361743 Central circular Delannoy numbers: a(n) is the number of Delannoy loops on an n X n toroidal grid.

1, 2, 16, 114, 768, 5010, 32016, 201698, 1257472, 7777314, 47800080, 292292946, 1779856128, 10799942322, 65336473104, 394246725570, 2373580947456, 14262064668738, 85546366040592, 512323096241714, 3063932437123840, 18300660294266322, 109183694129335056

Offset: 0

Author

Noah Snyder, Mar 22 2023

Keywords

• nonn

Comments

An (n,m) Delannoy loop is an oriented unbased loop on a toroidal grid with points labeled by Z/n x Z/m composed of steps of the form (1,0), (0,1), and (1,1), and which loops around the torus exactly once in each of the x-direction and the y-direction. The central circular Delannoy numbers count the number of (n,n) Delannoy loops. This is a modification of the ordinary central Delannoy numbers A001850.

Dimensions of endomorphism algebras End(S^{{n}}) in the circular Delannoy category attached to the oligomorphic group of order-preserving self-bijections of the circle.

Examples

			When n=2 see Figure 3 of "The circular Delannoy Category".
		

Links

• Winston de Greef, Table of n, a(n) for n = 0..1296
• Nate Harman, Andrew Snowden, and Noah Snyder, The circular Delannoy Category, arxiv: 2303.10814 [math.RT], 2023.

Crossrefs

Circular analog of A001850.

Main diagonal of A361745.

Cf. A008288, A108666, A002003, A047781.

Programs

Mathematica

a[n_Integer?Positive] := Sum[k Binomial[n, k] Binomial[n, k] 2^k, {k, 1, n}]

PARI

a(n) = if(n == 0, 1, sum(k=0, n, binomial(n, k)^2*k*2^k)) \\ Winston de Greef, Mar 22 2023

Python

from math import comb
def A361743(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

Formula

a(n) = Sum_{k=0..n} binomial(n,k)^2*k*2^k for n >= 1.

a(n) = n*(D(n,n-1) + D(n-1,n-1)) = n*(D(n,n) - D(n-1,n)) for n >= 1, where D(i,j) = A008288(i,j) are the Delannoy numbers.

a(n) = 2*A108666(n) for n >= 1.

From Alois P. Heinz, Mar 22 2023: (Start)

G.f.: 1 + 2*(1-x)*x/sqrt(x^2-6*x+1)^3.

a(n) = n*A002003(n) for n >= 1.

a(n) = 2*n*A047781(n) for n >= 1. (End)

a(n) = 2*n^2*hypergeom([1 - n, 1 - n], [2], 1) for n >= 1. - Peter Luschny, Mar 22 2023