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

%I A361743 #41 Mar 23 2023 05:09:03
%S A361743 1,2,16,114,768,5010,32016,201698,1257472,7777314,47800080,292292946,
%T A361743 1779856128,10799942322,65336473104,394246725570,2373580947456,
%U A361743 14262064668738,85546366040592,512323096241714,3063932437123840,18300660294266322,109183694129335056
%N A361743 Central circular Delannoy numbers: a(n) is the number of Delannoy loops on an n X n toroidal grid.
%C A361743 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.
%C A361743 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.
%H A361743 Winston de Greef, <a href="/A361743/b361743.txt">Table of n, a(n) for n = 0..1296</a>
%H A361743 Nate Harman, Andrew Snowden, and Noah Snyder, <a href="https://arxiv.org/abs/2303.10814">The circular Delannoy Category</a>, arxiv: 2303.10814 [math.RT], 2023.
%F A361743 a(n) = Sum_{k=0..n} binomial(n,k)^2*k*2^k for n >= 1.
%F A361743 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.
%F A361743 a(n) = 2*A108666(n) for n >= 1.
%F A361743 From _Alois P. Heinz_, Mar 22 2023: (Start)
%F A361743 G.f.: 1 + 2*(1-x)*x/sqrt(x^2-6*x+1)^3.
%F A361743 a(n) = n*A002003(n) for n >= 1.
%F A361743 a(n) = 2*n*A047781(n) for n >= 1. (End)
%F A361743 a(n) = 2*n^2*hypergeom([1 - n, 1 - n], [2], 1) for n >= 1. - _Peter Luschny_, Mar 22 2023
%e A361743 When n=2 see Figure 3 of "The circular Delannoy Category".
%t A361743 a[n_Integer?Positive] := Sum[k Binomial[n, k] Binomial[n, k] 2^k, {k, 1, n}]
%o A361743 (Python)
%o A361743 from math import comb
%o A361743 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
%o A361743 (PARI) a(n) = if(n == 0, 1, sum(k=0, n, binomial(n, k)^2*k*2^k)) \\ _Winston de Greef_, Mar 22 2023
%Y A361743 Circular analog of A001850.
%Y A361743 Main diagonal of A361745.
%Y A361743 Cf. A008288, A108666, A002003, A047781.
%K A361743 nonn
%O A361743 0,2
%A A361743 _Noah Snyder_, Mar 22 2023