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

%I A168288 #9 Oct 22 2018 10:34:58
%S A168288 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,
%T A168288 1389,177,1,1,367,4791,15263,15263,4791,367,1,1,749,15787,80285,
%U A168288 134647,80285,15787,749,1,1,1515,50529,393657,1033401,1033401,393657,50529
%N A168288 T(n,k) = 3*A046802(n+1,k+1) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).
%F A168288 E.g.f.: 3*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*exp(t*(1 + x)).
%e A168288 Triangle begins:
%e A168288      1;
%e A168288      1,   1;
%e A168288      1,   5,     1;
%e A168288      1,  15,    15,     1;
%e A168288      1,  37,    87,    37,      1;
%e A168288      1,  83,   373,   373,     83,     1;
%e A168288      1, 177,  1389,  2609,   1389,   177,     1;
%e A168288      1, 367,  4791, 15263,  15263,  4791,   367,   1;
%e A168288      1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1;
%e A168288       ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%t A168288 p[t_] = 3*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 2*Exp[t*(1 + x)];
%t A168288 Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten
%o A168288 (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)$
%o A168288 T(n, k) := 3*A046802(n + 1, k + 1) - 2*binomial(n, k)$
%o A168288 create_list(T(n, k), n, 0, 10, k, 0, n);
%o A168288 /* _Franck Maminirina Ramaharo_, Oct 21 2018 */
%Y A168288 Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y A168288 Cf. A142147, A142175, A168287, A168289, A168290, A168291, A168292, A168293.
%K A168288 nonn,easy,tabl
%O A168288 0,5
%A A168288 _Roger L. Bagula_, Nov 22 2009
%E A168288 Edited, and new name by _Franck Maminirina Ramaharo_, Oct 21 2018