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A168293 T(n,k) = 12*A046802(n+1,k+1) - 9*A008518(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

%I A168293 #8 Oct 22 2018 10:34:34
%S A168293 1,1,1,1,14,1,1,33,33,1,1,64,186,64,1,1,119,724,724,119,1,1,222,2415,
%T A168293 5120,2415,222,1,1,421,7491,28799,28799,7491,421,1,1,812,22456,142268,
%U A168293 257866,142268,22456,812,1,1,1587,66342,649554,1934544,1934544,649554
%N A168293 T(n,k) = 12*A046802(n+1,k+1) - 9*A008518(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).
%F A168293 E.g.f.: 12*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 9*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - 2*exp(t*(1 + x)).
%e A168293 Triangle begins:
%e A168293     1;
%e A168293     1,   1;
%e A168293     1,  14,     1;
%e A168293     1,  33,    33,      1;
%e A168293     1,  64,   186,     64,      1;
%e A168293     1, 119,   724,    724,    119,      1;
%e A168293     1, 222,  2415,   5120,   2415,    222,     1;
%e A168293     1, 421,  7491,  28799,  28799,   7491,   421,   1;
%e A168293     1, 812, 22456, 142268, 257866, 142268, 22456, 812, 1:
%e A168293      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%o A168293 (Maxima)
%o A168293 A123125(n, k) := sum((-1)^(k - j)*(binomial(n - j, k - j))*stirling2(n, j)*j!, j, 0, k)$
%o A168293 A046802(n, k) := sum(binomial(n - 1, r)*A123125(r, k - 1), r, k - 1, n - 1)$
%o A168293 A008518(n, k) := A123125(n, k) + A123125(n, k + 1)$
%o A168293 T(n, k) := 12*A046802(n + 1, k + 1) - 9*A008518(n, k) - 2*binomial(n, k)$
%o A168293 create_list(T(n, k), n, 0, 10, k, 0, n);
%o A168293 /* _Franck Maminirina Ramaharo_, Oct 21 2018 */
%Y A168293 Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y A168293 Cf. A142147, A142175, A168287, A168288, A168289, A168290, A168291, A168292.
%K A168293 nonn,tabl,easy,less
%O A168293 0,5
%A A168293 _Roger L. Bagula_, Nov 22 2009
%E A168293 Edited, and new name by _Franck Maminirina Ramaharo_, Oct 21 2018