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A333678
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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.
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4
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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
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OFFSET
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0,4
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LINKS
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MAPLE
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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);
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MATHEMATICA
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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]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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