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

%I A168289 #10 Oct 22 2018 10:34:50
%S A168289 1,1,1,1,6,1,1,19,19,1,1,48,114,48,1,1,109,494,494,109,1,1,234,1847,
%T A168289 3472,1847,234,1,1,487,6381,20339,20339,6381,487,1,1,996,21040,107028,
%U A168289 179506,107028,21040,996,1,1,2017,67360,524848,1377826,1377826,524848
%N A168289 T(n,k) = 4*A046802(n+1,k+1) - 3*A007318(n,k), triangle read by rows (0 <= k <= n).
%F A168289 E.g.f: 4*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 3*exp(t*(1 + x)).
%e A168289 Triangle begins:
%e A168289      1;
%e A168289      1,   1;
%e A168289      1,   6,     1;
%e A168289      1,  19,    19,      1;
%e A168289      1,  48,   114,     48,      1;
%e A168289      1, 109,   494,    494,    109,      1;
%e A168289      1, 234,  1847,   3472,   1847,    234,     1;
%e A168289      1, 487,  6381,  20339,  20339,   6381,   487,   1;
%e A168289      1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1;
%e A168289       ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%t A168289 p[t_] = 4*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 3*Exp[t*(1 + x)];
%t A168289 Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten
%o A168289 (Maxima)
%o A168289 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)$
%o A168289 T(n, k) := 4*A046802(n + 1, k + 1) - 3*binomial(n, k)$
%o A168289 create_list(T(n, k), n, 0, 10, k, 0, n);
%o A168289 /* _Franck Maminirina Ramaharo_, Oct 21 2018 */
%Y A168289 Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y A168289 Cf. A142147, A142175, A168287, A168288, A168290, A168291, A168292, A168293.
%K A168289 nonn,easy,tabl
%O A168289 0,5
%A A168289 _Roger L. Bagula_, Nov 22 2009
%E A168289 Edited, and new name by _Franck Maminirina Ramaharo_, Oct 21 2018