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Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292.
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%I #38 Jun 25 2021 13:07:48

%S 1,1,2,1,8,3,1,22,33,4,1,52,198,104,5,1,114,906,1208,285,6,1,240,3573,

%T 9664,5955,720,7,1,494,12879,62476,78095,25758,1729,8,1,1004,43824,

%U 352936,780950,529404,102256,4016,9,1,2026,143520,1820768,6551770,7862124,3186344,382720,9117,10

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

%C 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.

%C 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

%H Alois P. Heinz, <a href="/A065826/b065826.txt">Rows n = 1..141, flattened</a>

%H Ron M. Adin, Sergi Elizalde, Victor Reiner, Yuval Roichman, <a href="http://ceur-ws.org/Vol-2113/paper2.pdf">Cyclic descent extensions and distributions</a>, Semantic Sensor Networks Workshop 2018, CEUR Workshop Proceedings (2018) Vol. 2113.

%H Ron M. Adin, Victor Reiner, Yuval Roichman, <a href="https://arxiv.org/abs/1710.06664">On cyclic descents for tableaux</a>, arXiv:1710.06664 [math.CO], 2017.

%F 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).

%F 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

%e Rows start:

%e 1;

%e 1, 2;

%e 1, 8, 3;

%e 1, 22, 33, 4;

%e 1, 52, 198, 104, 5;

%e 1, 114, 906, 1208, 285, 6;

%e 1, 240, 3573, 9664, 5955, 720, 7;

%e 1, 494, 12879, 62476, 78095, 25758, 1729, 8;

%e etc.

%p 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

%t Array[Range[Length@ #] # &@ CoefficientList[(1 - x)^(# + 1)*PolyLog[-#, x]/x, x] &, 10] (* _Michael De Vlieger_, Sep 24 2018, after _Vaclav Kotesovec_ at A008292 *)

%o (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

%Y Cf. A008292.

%K nonn,tabl

%O 1,3

%A _Henry Bottomley_, Dec 06 2001