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
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Dec 06 2001
STATUS
approved