A274373 Sum of the areas of all modified skew Dyck paths of semilength n.

0, 1, 6, 35, 188, 989, 5131, 26411, 135229, 689814, 3509014, 17811637, 90256685, 456719880, 2308440442, 11656409995, 58809893357, 296500180806, 1493924791698, 7523064390774, 37866103978109, 190510720248534, 958122016323881, 4816944544836927, 24209532464417585

Offset: 0

Author

Alois P. Heinz, Jun 19 2016

Keywords

• nonn

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

Examples

			a(3) = 35 = 9+7+5+6+5+3 = sum of the areas of UUUDDD, UUDUDD, UUDDUD, UAUDDD, UDUUDD, UDUDUD, respectively.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A230823, A274372.

Programs

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 *)

Formula

a(n) = Sum_{k=n..n^2} k * A274372(n,k).