A065826 Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292.

1, 1, 2, 1, 8, 3, 1, 22, 33, 4, 1, 52, 198, 104, 5, 1, 114, 906, 1208, 285, 6, 1, 240, 3573, 9664, 5955, 720, 7, 1, 494, 12879, 62476, 78095, 25758, 1729, 8, 1, 1004, 43824, 352936, 780950, 529404, 102256, 4016, 9, 1, 2026, 143520, 1820768, 6551770, 7862124, 3186344, 382720, 9117, 10

Offset: 1

Author

Henry Bottomley, Dec 06 2001

Keywords

• nonn
• tabl

Comments

Row sums are (n+1)!/2, i.e., A001710 offset, implying that if n balls are put at random into n boxes, the expected number of boxes with at least one ball is (n+1)/2 and the expected number of empty boxes is (n-1)/2.

T(n,k) is the number of permutations of {1,2,...,n+1} that start with an ascent and that have k-1 descents. - Ira M. Gessel, May 02 2017

Examples

			Rows start:
  1;
  1,   2;
  1,   8,     3;
  1,  22,    33,     4;
  1,  52,   198,   104,     5;
  1, 114,   906,  1208,   285,     6;
  1, 240,  3573,  9664,  5955,   720,    7;
  1, 494, 12879, 62476, 78095, 25758, 1729, 8;
  etc.
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Ron M. Adin, Sergi Elizalde, Victor Reiner, Yuval Roichman, Cyclic descent extensions and distributions, Semantic Sensor Networks Workshop 2018, CEUR Workshop Proceedings (2018) Vol. 2113.
• Ron M. Adin, Victor Reiner, Yuval Roichman, On cyclic descents for tableaux, arXiv:1710.06664 [math.CO], 2017.

Crossrefs

Cf. A008292.

Programs

GAP

Flat(List([1..10],n->List([1..n],k->Sum([0..k],j->k*(-1)^j*(k-j)^n*Binomial(n+1,j))))); # Muniru A Asiru, Mar 09 2019

Maple

T:=(n,k)->add(k*(-1)^j*(k-j)^n*binomial(n+1,j),j=0..k): seq(seq(T(n,k),k=1..n),n=1..10); # Muniru A Asiru, Mar 09 2019

Mathematica

Array[Range[Length@ #] # &@ CoefficientList[(1 - x)^(# + 1)*PolyLog[-#, x]/x, x] &, 10] (* Michael De Vlieger, Sep 24 2018, after Vaclav Kotesovec at A008292 *)

Formula

T(n, k) = k*(a(n-1, k) + a(n-1, k-1)*(n-k+1)/(k-1)) [with T(n, 1) = 1] = Sum_{j=0..k} k*(-1)^j*(k-j)^n*binomial(n+1, j).

E.g.f.: (exp(x*(1-t)) - 1 - x*(1-t))/((1-t)*(1 - t*exp(x*(1-t)))). - Ira M. Gessel, May 02 2017