cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Noah Snyder, Mar 22 2023

Keywords

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

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

A361758 a(n) = [x^n] (x^5 + 5*x^4 + 4*x^3 - 3*x + 1)/((1 - x)*(x^2 + 2*x - 1)^2).

Original entry on oeis.org

1, 2, 4, 10, 30, 90, 264, 754, 2106, 5778, 15628, 41786, 110678, 290858, 759312, 1971042, 5091442, 13095586, 33555988, 85695978, 218198158, 554081146, 1403588376, 3547702610, 8949110954, 22532603954, 56637795100, 142141826074, 356212187334, 891481312842
Offset: 0

Views

Author

Peter Luschny, Mar 23 2023

Keywords

Crossrefs

Cf. A361745.

Programs

  • Maple
    a := proc(n) option remember; if n < 4 then return [1, 2, 4, 10][n + 1] fi;
((3*n^2 - 7*n + 2)*a(n - 1) - (n^2 - n)*a(n - 3) - (n^2 - 3*n)*a(n - 2))/((n - 1)*(n - 2)) end: seq(a(n), n = 0..29);
# Alternative:
ogf := (x^5 + 5*x^4 + 4*x^3 - 3*x + 1)/((1 - x)*(x^2 + 2*x - 1)^2):
ser := series(ogf, x, 40): seq(coeff(ser, x, n), n = 0..29);

Formula

a(n) = ((3*n^2 - 7*n + 2)*a(n - 1) - (n^2 - n)*a(n - 3) - (n^2 - 3*n)*a(n - 2)) / ((n - 1)*(n - 2)) for n >= 4.
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