A229078 Number of ascending runs in {1,...,n}^n.

0, 1, 7, 63, 736, 10625, 182736, 3647119, 82837504, 2109289329, 59500000000, 1841557146671, 62041198952448, 2259914256880657, 88499197217837056, 3707501605224609375, 165444235911082541056, 7834451891982365825441, 392371124973096027488256

Offset: 0

Author

Alois P. Heinz, Sep 12 2013

Keywords

• nonn

Examples

			a(1) = 1: [1].
a(2) = 7 = 2+2+1+2: [1,1], [2,1], [1,2], [2,2].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Lambert W-Function
• Wikipedia, Lambert W function

Crossrefs

Main diagonal of A229079.

Cf. A062023 (nondescending runs), A066274.

Programs

Maple

a:= n-> `if`(n=0, 0, n^(n-1)*(n*(n+2)-1)/2):
seq(a(n), n=0..25);

Formula

a(n) = n^(n-1)*(n*(n+2)-1)/2 for n>0, a(0) = 0.

E.g.f.: 1/2*W(-x)*(W(-x)^3+W(-x)^2-W(-x)-2)/(1+W(-x))^3, W(x) Lambert's function (principal branch).

a(n) = A062023(n) + A066274(n) for n>0.