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A333071
Total area under all lattice paths from (0,0) to (n,0) that do not go below the x-axis, and at (x,y) only allow steps (1,v) with v in {-1,0,1,...,y+1}.
5
0, 0, 1, 4, 16, 63, 239, 895, 3343, 12503, 46905, 176620, 667664, 2533699, 9650737, 36887383, 141448958, 544022417, 2098082719, 8111788699, 31434420426, 122068414186, 474932563378, 1851059631879, 7226108097869, 28250493771358, 110594307388370, 433488248791630
OFFSET
0,4
LINKS
MAPLE
b:= proc(x, y) option remember; `if`(x=0, [1, 0],
add(`if`(x+j>y, (p-> p+[0, p[1]*(y-j/2)])(
b(x-1, y-j)), 0), j=-1-y..min(1, y)))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);
MATHEMATICA
b[x_, y_] := b[x, y] = If[x == 0, {1, 0},
Sum[If[x + j > y, With[{p = b[x - 1, y - j]}, p +
{0, p[[1]] (y - j/2)}], 0], {j, -1 - y, Min[1, y]}]];
a[n_] := b[n, 0][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 05 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Mar 06 2020
STATUS
approved