A168291 T(n,k) = 4*A046802(n+1,k+1) - 2*A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 32, 82, 32, 1, 1, 65, 330, 330, 65, 1, 1, 130, 1159, 2304, 1159, 130, 1, 1, 259, 3801, 13195, 13195, 3801, 259, 1, 1, 516, 12016, 67316, 117170, 67316, 12016, 516, 1, 1, 1029, 37212, 319332, 889230, 889230, 319332, 37212

Offset: 0

Author

Roger L. Bagula, Nov 22 2009

Keywords

• nonn
• easy
• less
• tabl

Examples

			Triangle begins:
     1;
     1,   1;
     1,   6,     1;
     1,  15,    15,     1;
     1,  32,    82,    32,      1;
     1,  65,   330,   330,     65,     1;
     1, 130,  1159,  2304,   1159,   130,     1;
     1, 259,  3801, 13195,  13195,  3801,   259,   1;
     1, 516, 12016, 67316, 117170, 67316, 12016, 516, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
		

Crossrefs

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168289, A168290, A168292, A168293.

Programs

Maxima

A123125(n, k) := sum((-1)^(k - j)*(binomial(n - j, k - j))*stirling2(n, j)*j!, j, 0, k)$
A046802(n, k) := sum(binomial(n - 1, r)*A123125(r, k - 1), r, k - 1, n - 1)$
A008518(n, k) := A123125(n, k) + A123125(n, k + 1)$
T(n, k) := 4*A046802(n + 1, k + 1) - 2*A008518(n, k) - binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

Formula

E.g.f.: 4*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - exp(t*(1 + x)).

Extensions

Edited, new name by Franck Maminirina Ramaharo, Oct 21 2018