A168287 T(n,k) = 2*A046802(n+1,k+1) - A007318(n,k), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 60, 26, 1, 1, 57, 252, 252, 57, 1, 1, 120, 931, 1746, 931, 120, 1, 1, 247, 3201, 10187, 10187, 3201, 247, 1, 1, 502, 10534, 53542, 89788, 53542, 10534, 502, 1, 1, 1013, 33698, 262466, 688976, 688976, 262466, 33698, 1013

Offset: 0

Author

Roger L. Bagula, Nov 22 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins:
     1;
     1,    1;
     1,    4,     1;
     1,   11,    11,     1;
     1,   26,    60,    26,     1;
     1,   57,   252,   252,    57,     1;
     1,  120,   931,  1746,   931,   120,     1;
     1,  247,  3201, 10187, 10187,  3201,   247,   1;
     1,  502, 10534, 53542, 89788, 53542, 10534, 502, 1;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
		

Crossrefs

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

Cf. A142147, A142175, A168288, A168289, A168290, A168291, A168292, A168293.

Programs

Mathematica

p[t_] = 2*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 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) := 2*A046802(n + 1, k + 1) - binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

Formula

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

Extensions

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