OFFSET
0,4
LINKS
Eric Weisstein's World of Mathematics, Eulerian Number
Wikipedia, Eulerian number
FORMULA
Sum_{k=0..n} T(n,k) = A005096(n), n > 0.
From Franck Maminirina Ramaharo, Oct 06 2018: (Start)
T(n,k) = (2*k - n)*Sum_{j=0..k} (-1)^j*(k - j + 1)^n*binomial(n + 1, j) for 0 <= k <= n - 1 and T(n,n) = n.
T(2*n-1,n-1) = -A025585(n).
T(2*n,n-1) = -A177042(n). (End)
EXAMPLE
Triangle begins:
0;
-1, 1;
-2, 0, 2;
-3, -4, 1, 3;
-4, -22, 0, 2, 4;
-5, -78, -66, 26, 3, 5;
-6, -228, -604, 0, 114, 4, 6;
-7, -600, -3573, -2416, 1191, 360, 5, 7;
-8, -1482, -17172, -31238, 0, 8586, 988, 6, 8;
-9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9;
...
MAPLE
T:= proc(n, k) `if`(n=k, n, (2*k-n)*add((-1)^j*(k-j+1)^n*binomial(n+1, j), j=0..k)); end proc: seq(seq(T(n, k), k=0..n), n=0..10); # Muniru A Asiru, Oct 06 2018
T := (n, k) -> `if`(n = k, n, (2*k - n)*combinat:-eulerian1(n, k)):
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Oct 06 2018
MATHEMATICA
T[n_, k_] = If[n == k, n, (2*k - n)*Sum[(-1)^j*(k - j + 1)^n*Binomial[n + 1, j], {j, 0, k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}]//Flatten
PROG
(Maxima) T(n, k) := if n = k then n else (2*k - n)*sum((-1)^j*(k - j + 1)^n*binomial(n + 1, j), j, 0, k)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 05 2018 */
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Sep 09 2008
EXTENSIONS
Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 06 2018
STATUS
approved