A048116 -id:A048116 - cp's OEIS frontend

A048113 Triangular array T read by rows: T(h,k) = number of paths consisting of steps from (1,1) to (h,k) such that each step has length 1 directed up or right and each vertex (i,j) satisfies i/2<=j<=2i, for h=0,1,2,... and k=0,1,2,...

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 6, 12, 6, 0, 0, 0, 0, 0, 0, 6, 18, 18, 6, 0, 0, 0, 0, 0, 0, 0, 24, 36, 24, 0, 0, 0, 0, 0, 0, 0, 0, 24, 60, 60, 24, 0, 0, 0, 0, 0, 0, 0, 0, 24, 84, 120, 84, 24, 0, 0, 0, 0

Offset: 0

Clark Kimberling

			Rows: {0}; {0,0}; {0,1,0}; {0,1,1,0}; ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A048116.

T[0, 0] = T[0, 1] = T[1, 0] = 0; T[1, 1] = T[2, 1] = T[1, 2] = 1; T[h_, k_] /; Not[h/2 <= k <= 2h] = 0; T[h_, k_] := T[h, k] = If[h-1 >= k/2, T[h-1, k], 0] + If[k-1 >= h/2, T[h, k-1], 0]; row[s_] := Table[T[h, s-h], {h, 0, s}]; Table[row[s], {s, 0, 12}] // Flatten (* Jean-François Alcover, Dec 02 2016 *)

Offset changed to 0 by Alois P. Heinz, Oct 06 2016

A277262 Number of walks on cubic lattice starting at (1,1,1), ending at (n,n,n), remaining in the first (nonnegative) octant and using steps (0,-1,2), (0,2,-1), (-1,0,2), (2,0,-1), (-1,2,0), and (2,-1,0).

Original entry on oeis.org

0, 1, 12, 456, 54216, 6932916, 1069256400, 170663949024, 29130191148240, 5115288488816760, 927446504770571520, 171486284915686699620, 32295496327107026335392, 6164943698859825359296740, 1190940852937573264531168944, 232287567721717805821704554232

Offset: 0

Alois P. Heinz, Oct 07 2016

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A048116.

g():= combinat[permute]([0, -1, 2]):
b:= proc(l) option remember; `if`(l=[1$3], 1, add((p->
      `if`(p[1]<0, 0, b(p)))(sort(l-x)), x=g()))
    end:
a:= n-> b([n$3]):
seq(a(n), n=0..20);

g = Permutations[{0, -1, 2}];
b[l_] := b[l] = If[l == {1, 1, 1}, 1, Sum[Function[p, If[p[[1]] < 0, 0, b[p]]][Sort[l - x]], {x, g}]];
a[n_] := b[{n, n, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)

a(n) ~ c * 6^(3*n) / n, where c = 0.000020280187096503586851533... . - Vaclav Kotesovec, Oct 14 2016

A277248 Number of planar walks starting at (1,1), ending at (3n,0), remaining in the first quadrant and using steps (-1,2) and (2,-1).

Original entry on oeis.org

1, 2, 6, 24, 108, 528, 2724, 14616, 80760, 456552, 2628504, 15360216, 90879096, 543336912, 3277586136, 19924733088, 121943223576, 750756116376, 4646484480552, 28892787031008, 180420486241776, 1130930538186360, 7113550964713848, 44885329202906448

Offset: 1

Feng Jishe, Oct 06 2016

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Bousquet-Mélou, M. Petkovsek, Walks confined in a quadrant are not always D-finite, Theoretical Computer Science, 307(2003): 257-276.
Ira M. Gessel, A probabilistic method for lattice path enumeration, Journal of statistical planning and inference, 14 (1986), 49-58.

Cf. A048116.

b:= proc(l) option remember; `if`(l=[1$2], 1, add((p->
      `if`(p[1]<0, 0, b(p)))(sort((l-x))), x=[[-1, 2], [2, -1]]))
    end:
a:= n-> b([0,3*n]):
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 06 2016

b[l_List] := b[l] = If[l == {1, 1}, 1, Sum[Function[p, If[p[[1]]<0, 0, b[p]]][Sort[l-x]], {x, {{-1, 2}, {2, -1}}}]]; a[n_] := b[{0, 3n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Dec 04 2016 after Alois P. Heinz *)

a(n) ~ c * (27/4)^n / n^(3/2), where c = 0.06045583689606517807688682344735167414726208387456561322459238109992522838... . - Vaclav Kotesovec, Oct 07 2016

More terms from Alois P. Heinz, Oct 06 2016

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A048113 Triangular array T read by rows: T(h,k) = number of paths consisting of steps from (1,1) to (h,k) such that each step has length 1 directed up or right and each vertex (i,j) satisfies i/2<=j<=2i, for h=0,1,2,... and k=0,1,2,...

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A277262 Number of walks on cubic lattice starting at (1,1,1), ending at (n,n,n), remaining in the first (nonnegative) octant and using steps (0,-1,2), (0,2,-1), (-1,0,2), (2,0,-1), (-1,2,0), and (2,-1,0).

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A277248 Number of planar walks starting at (1,1), ending at (3n,0), remaining in the first quadrant and using steps (-1,2) and (2,-1).

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