A141693 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.

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, -10

Offset

0,4

Links

Table of n, a(n) for n=0..55.

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

Cf. A008292.

Sequence in context: A284268 A063180 A263624 * A279679 A261096 A257092

Adjacent sequences: A141690 A141691 A141692 * A141694 A141695 A141696

Keyword

tabl,sign,easy

Author

Roger L. Bagula, Sep 09 2008

Extensions

Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 06 2018

Status

approved