A267862 Number of planar lattice convex polygonal lines joining the origin and the point (n,n).

1, 2, 5, 13, 32, 77, 178, 399, 877, 1882, 3959, 8179, 16636, 33333, 65894, 128633, 248169, 473585, 894573, 1673704, 3103334, 5705383, 10405080, 18831761, 33836627, 60378964, 107035022, 188553965, 330166814, 574815804, 995229598, 1714004131, 2936857097

Offset: 0

Author

Christoph Koutschan, Apr 07 2016

Keywords

• nonn
• walk

Comments

In other words, we are counting walks on the integer lattice N^2 that start at (0,0) and end at (n,n); they may take arbitrary steps, but the slopes of the steps in the walk must strictly increase. As a result, we obtain a convex polygon when joining the two endpoints of the walk with the point (0,n).

Examples

			The two walks for n = 1 are
(0,0) -> (1,1)
(0,0) -> (1,0) -> (1,1).
The five possibilities for n = 2 are
(0,0) -> (2,2)
(0,0) -> (1,0) -> (2,1) -> (2,2)
(0,0) -> (1,0) -> (2,2)
(0,0) -> (2,0) -> (2,2)
(0,0) -> (2,1) -> (2,2).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100
• J. Bureaux, N. Enriquez, On the number of lattice convex chains, arXiv:1603.09587 [math.PR], 2016.

Crossrefs

Cf. A002774, A090806, A219554.

Programs

Mathematica

a[i_Integer, j_Integer, s_] := a[i, j, s] = If[i === 0, 1, Sum[a[i - x, j - y, y/x], {x, 1, i}, {y, Floor[s*x] + 1, j}]]; a[n_Integer] := a[n] = 1 + Sum[a[n - x, n - y, y/x], {x, 1, n}, {y, 0, x - 1}]; Flatten[{1, Table[a[n], {n, 30}]}]
nmax = 20; p = (1 - x)*(1 - y); Do[Do[p = Expand[p*If[GCD[i, j] == 1, (1 - x^i*y^j), 1]]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {i, 1, nmax}], {j, 1, nmax}]; p = Expand[Normal[Series[1/p, {x, 0, nmax}, {y, 0, nmax}]]]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Flatten[{1, Table[Coefficient[p, x^n*y^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 08 2016 *)

Formula

a(n) = [x^n*y^n] 1/((1-x)*(1-y)*Product_{i>0,j>0,gcd(i,j)=1} (1-x^i*y^j)).

An asymptotic formula for a(n) is given by Bureaux and Enriquez: a(n) ~ e^(-2*zeta'(-1))/((2*Pi)^(7/6)*sqrt(3)*kappa^(1/18)*n^(17/18)) * e^(3*kappa^(1/3)*n^(2/3)+...) where kappa := zeta(3)/zeta(2) and zeta denotes the Riemann zeta function.