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%I #19 Oct 06 2018 03:58:50
%S 0,-1,1,-2,0,2,-3,-4,1,3,-4,-22,0,2,4,-5,-78,-66,26,3,5,-6,-228,-604,
%T 0,114,4,6,-7,-600,-3573,-2416,1191,360,5,7,-8,-1482,-17172,-31238,0,
%U 8586,988,6,8,-9,-3514,-73040,-264702,-156190,88234,43824,2510,7,9,-10
%N Triangle read by rows: T(n,k) = (2*k - n)*A008292(n,k) with T(n,n) = n, 0 <= k <= n, where A008292 is the triangle of Eulerian numbers.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eulerian_number">Eulerian number</a>
%F Sum_{k=0..n} T(n,k) = A005096(n), n > 0.
%F From _Franck Maminirina Ramaharo_, Oct 06 2018: (Start)
%F 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.
%F T(2*n-1,n-1) = -A025585(n).
%F T(2*n,n-1) = -A177042(n). (End)
%e Triangle begins:
%e 0;
%e -1, 1;
%e -2, 0, 2;
%e -3, -4, 1, 3;
%e -4, -22, 0, 2, 4;
%e -5, -78, -66, 26, 3, 5;
%e -6, -228, -604, 0, 114, 4, 6;
%e -7, -600, -3573, -2416, 1191, 360, 5, 7;
%e -8, -1482, -17172, -31238, 0, 8586, 988, 6, 8;
%e -9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9;
%e ...
%p 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
%p T := (n, k) -> `if`(n = k, n, (2*k - n)*combinat:-eulerian1(n,k)):
%p seq(seq(T(n,k), k=0..n), n=0..9); # _Peter Luschny_, Oct 06 2018
%t T[n_, k_] = If[n == k, n, (2*k - n)*Sum[(-1)^j*(k - j + 1)^n*Binomial[n + 1, j], {j, 0, k}]];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}]//Flatten
%o (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)$
%o tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* _Franck Maminirina Ramaharo_, Oct 05 2018 */
%Y Cf. A008292.
%K tabl,sign,easy
%O 0,4
%A _Roger L. Bagula_, Sep 09 2008
%E Edited, new name and offset corrected by _Franck Maminirina Ramaharo_, Oct 06 2018
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