A168289 T(n,k) = 4*A046802(n+1,k+1) - 3*A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 48, 114, 48, 1, 1, 109, 494, 494, 109, 1, 1, 234, 1847, 3472, 1847, 234, 1, 1, 487, 6381, 20339, 20339, 6381, 487, 1, 1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1, 1, 2017, 67360, 524848, 1377826, 1377826, 524848

Offset: 0

Author

Roger L. Bagula, Nov 22 2009

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins:
     1;
     1,   1;
     1,   6,     1;
     1,  19,    19,      1;
     1,  48,   114,     48,      1;
     1, 109,   494,    494,    109,      1;
     1, 234,  1847,   3472,   1847,    234,     1;
     1, 487,  6381,  20339,  20339,   6381,   487,   1;
     1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
		

Crossrefs

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

Cf. A142147, A142175, A168287, A168288, A168290, A168291, A168292, A168293.

Programs

Mathematica

p[t_] = 4*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 3*Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

Maxima

A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 4*A046802(n + 1, k + 1) - 3*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))) - 3*exp(t*(1 + x)).

Extensions

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