A168293 T(n,k) = 12*A046802(n+1,k+1) - 9*A008518(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 14, 1, 1, 33, 33, 1, 1, 64, 186, 64, 1, 1, 119, 724, 724, 119, 1, 1, 222, 2415, 5120, 2415, 222, 1, 1, 421, 7491, 28799, 28799, 7491, 421, 1, 1, 812, 22456, 142268, 257866, 142268, 22456, 812, 1, 1, 1587, 66342, 649554, 1934544, 1934544, 649554

Offset: 0

Author

Roger L. Bagula, Nov 22 2009

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins:
    1;
    1,   1;
    1,  14,     1;
    1,  33,    33,      1;
    1,  64,   186,     64,      1;
    1, 119,   724,    724,    119,      1;
    1, 222,  2415,   5120,   2415,    222,     1;
    1, 421,  7491,  28799,  28799,   7491,   421,   1;
    1, 812, 22456, 142268, 257866, 142268, 22456, 812, 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, A168291, A168292.

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) := 12*A046802(n + 1, k + 1) - 9*A008518(n, k) - 2*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

Formula

E.g.f.: 12*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 9*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - 2*exp(t*(1 + x)).

Extensions

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