OFFSET
0,2
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.
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.
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
EXAMPLE
When n=2 see Figure 3 of "The circular Delannoy Category".
MATHEMATICA
a[n_Integer?Positive] := Sum[k Binomial[n, k] Binomial[n, k] 2^k, {k, 1, n}]
PROG
(Python)
from math import comb
def A361743(n): return sum(comb(n, k)**2*k<<k-1 for k in range(1, n+1))<<1 if n else 1 # Chai Wah Wu, Mar 22 2023
(PARI) a(n) = if(n == 0, 1, sum(k=0, n, binomial(n, k)^2*k*2^k)) \\ Winston de Greef, Mar 22 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Noah Snyder, Mar 22 2023
STATUS
approved