A083385 Total height of all elements in all preferential arrangements of n elements, where elements at the bottom level have height 1.

1, 8, 66, 616, 6530, 77964, 1037974, 15266192, 246003354, 4312200340, 81714050462, 1664849747928, 36296654286178, 843235868819036, 20797267023597030, 542745686844469024, 14942598078715420202, 432842048921633654052, 13159824571927634917678, 419012563973742290424680

Offset: 1

Author

N. J. A. Sloane, Jun 07 2003

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Crossrefs

Equals A083384(n) + n*A000670(n).

Cf. A000670, A083410, A261781.

Programs

Mathematica

a[n_] := n Sum[1/2 (k-1) k! StirlingS2[n, k], {k, 1, n}] + n Sum[(-1)^(k-j) Binomial[k, j] j^n, {j, 0, n}, {k, 0, n}];
Array[a, 20] (* Jean-François Alcover, Sep 01 2018 *)

Formula

See A083384 for formula.

a(n) = A261781(n+1,n)/2. - Alois P. Heinz, Aug 10 2016

a(n) ~ n! * n^2 / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018