A168288 T(n,k) = 3*A046802(n+1,k+1) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 87, 37, 1, 1, 83, 373, 373, 83, 1, 1, 177, 1389, 2609, 1389, 177, 1, 1, 367, 4791, 15263, 15263, 4791, 367, 1, 1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1, 1, 1515, 50529, 393657, 1033401, 1033401, 393657, 50529

Offset: 0

Author

Roger L. Bagula, Nov 22 2009

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins:
     1;
     1,   1;
     1,   5,     1;
     1,  15,    15,     1;
     1,  37,    87,    37,      1;
     1,  83,   373,   373,     83,     1;
     1, 177,  1389,  2609,   1389,   177,     1;
     1, 367,  4791, 15263,  15263,  4791,   367,   1;
     1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
		

Crossrefs

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

Cf. A142147, A142175, A168287, A168289, A168290, A168291, A168292, A168293.

Programs

Mathematica

p[t_] = 3*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 2*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) := 3*A046802(n + 1, k + 1) - 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.: 3*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*exp(t*(1 + x)).

Extensions

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