A228959 Total sum of squared lengths of ascending runs in all permutations of [n].

0, 1, 6, 32, 186, 1222, 9086, 75882, 705298, 7231862, 81160422, 990024466, 13047411482, 184788881838, 2799459801742, 45178128866282, 773829771302946, 14021761172671462, 267991492197471158, 5388234382450264002, 113692608262971520042, 2512031106415692960926

Offset: 0

Author

Alois P. Heinz, Sep 09 2013

Keywords

• nonn

Examples

			a(0) = 0: ().
a(1) = 1: (1).
a(2) = 6 = 4+2: (1,2), (2,1).
a(3) = 32 = 9+5+5+5+5+3: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column k=2 of A229001.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, n*(2*n-1),
      (2*n+1)*a(n-1) -(n-1)*((n+2)*a(n-2)-(n-2)*a(n-3)))
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := With[{k = 2}, Sum[If[n==t, 1, (n!/(t+1)!)(t(n-t+1)+1-((t+1)(n-t)+1)/(t+2))] t^k, {t, 1, n}]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz in A229001 *)

Formula

E.g.f.: (2*exp(x)-x-2)/(x-1)^2.

a(n) = (2*n+1)*a(n-1)-(n-1)*((n+2)*a(n-2)-(n-2)*a(n-3)) for n>=3, a(n) = n*(2*n-1) for n<3.

a(n) ~ n! * (2*exp(1)-3)*n. - Vaclav Kotesovec, Sep 12 2013