OFFSET
0,3
COMMENTS
A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.
a(n)^(1/n) tends to 5. - Vaclav Kotesovec, Jun 26 2016
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum_{k=n..n^2} k * A274372(n,k).
EXAMPLE
a(3) = 35 = 9+7+5+6+5+3 = sum of the areas of UUUDDD, UUDUDD, UUDDUD, UAUDDD, UDUUDD, UDUDUD, respectively.
MAPLE
b:= proc(x, y, t, n) option remember; `if`(y>n, 0, `if`(n=y,
`if`(t=2, 0, [1, 0]), (p-> p+[0, p[1]*(2*y+1)])(b(x+1, y
+1, 0, n-1))+`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 0)
+`if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0)))
end:
a:= n-> b(0$3, 2*n)[2]:
seq(a(n), n=0..30);
MATHEMATICA
b[x_, y_, t_, n_] := b[x, y, t, n] = If[y > n, 0, If[n == y, If[t == 2, {0, 0}, {1, 0}], Function[p, p + {0, p[[1]] (2y + 1)}][b[x + 1, y + 1, 0, n - 1]] + If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1], 0] + If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]];
a[n_] := b[0, 0, 0, 2 n][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 19 2016
STATUS
approved