OFFSET
1,5
COMMENTS
Sum of row n = (n-1)!.
LINKS
Hsien-Kuei Hwang, Hua-Huai Chern, Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018.
FORMULA
T(n,k)= (k+ floor((-1)^(n-1)/2))*T(n-1,k) + (n-k- floor((-1)^(n-1)/2))*T(n-1,k-1), n=>2, 1<=k<=n, T(1,1)=1, T(n,0)=0, T(n,n+1)=0.
EXAMPLE
Triangle T(n,k) begins:
n\k 1 2 3 4 5 6 7 8
1 1
2 0 1
3 0 2 0
4 0 2 4 0
5 0 4 16 4 0
6 0 4 48 60 8 0
7 0 8 160 384 160 8 0
8 0 8 368 1952 2176 520 16 0
T(4,3)=4 [top 14,23,56,78; bottom 18,27,36,45] [top 16,25,34,78; bottom 18,27,36,45] [top 12,34,58,67; bottom 18,27,36,45] [top 12,38,47,56; bottom 18,27,36,45]
MATHEMATICA
T[1, 1]:= 1; T[n_, 0]:= 0; T[n_, n_+1] := 0; T[n_, k_]:= If[k == n+1, 0, (k + Floor[(-1)^(n-1)/2])*T[n-1, k] + (n-k -Floor[(-1)^(n-1)/2]) T[n-1, k - 1]]; Table[T[n, k], {n, 1, 15}, {k, 1, n}]//Flatten (* G. C. Greubel, Oct 10 2018 *)
PROG
(PARI) T(n, k)=if(n==1 && k==1, 1, if(k==0, 0, if( k==n+1, 0, (k+ floor((-1)^(n-1)/2))*T(n-1, k) + (n-k- floor((-1)^(n-1)/2))*T(n-1, k-1))));
for(n=1, 15, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 10 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger Ford, Jul 02 2014
STATUS
approved