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

1,3

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

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.

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A008292.

Sequence in context: A001281 A331312 A232535 * A286899 A123235 A176052

Adjacent sequences: A065823 A065824 A065825 * A065827 A065828 A065829

Keyword

nonn,tabl

Author

Henry Bottomley, Dec 06 2001

Status

approved