A247748 Total area below all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3).

0, 0, 1, 4, 29, 128, 683, 3048, 15631, 73584, 367878, 1766272, 8772765, 42778568, 212026492, 1041894304, 5164955809, 25511089124, 126587331646, 627213797064, 3115744498691, 15471098709336, 76936461658184, 382598697899276, 1904401873987964, 9480893249387192

Offset: 0

Author

Alois P. Heinz, Sep 23 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..700

Crossrefs

Cf. A240904, A247749.

Programs

Maple

b:= proc(x, y) option remember; `if`(y<0 or xp+[0, p[1]*(y+j/2)])
       (b(x-1, y+j)), j=[-1, -3, 0, 1, 3])))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);

Mathematica

b[x_, y_] := b[x, y] = If[y < 0 || x < y, {0, 0},
     If[x == 0, {1, 0}, Sum[Function[p, p + {0, p[[1]]*(y + j/2)}][
     b[x - 1, y + j]], {j, {-1, -3, 0, 1, 3}}]]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>=1} k * A247749(n,k).