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A333678
Total area under all nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.
4
0, 0, 0, 2, 7, 22, 64, 196, 574, 1762, 5379, 16378, 49380, 148892, 449004, 1353718, 4076150, 12267160, 36903433, 110979048, 333628384, 1002722482, 3013085711, 9052404522, 27192329061, 81671691634, 245271884478, 736513920180, 2211445194899, 6639545054310
OFFSET
0,4
MAPLE
b:= proc(x, y, t) option remember; `if`(x=0, [1, 0],
add((p-> p+[0, p[1]*(y+j/2)])(b(x-1, y+j, j)),
j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..38);
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[x == 0, {1, 0},
Sum[Function[p, p + {0, If[p === 0, 0, p[[1]]]*(y + j/2)}][
b[x-1, y+j, j]], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
a[n_] := b[n, 0, 0][[2]];
a /@ Range[0, 38] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 01 2020
STATUS
approved