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A168292 T(n,k) = 24*A046802(n+1,k+1) - 9*A008518(n,k) - 8*A007318(n,k), triangle read by rows (0 <= k <= n).

%I A168292 #12 Oct 22 2018 10:34:25
%S A168292 7,7,7,7,38,7,7,99,99,7,7,220,546,220,7,7,461,2236,2236,461,7,7,942,
%T A168292 8001,15596,8001,942,7,7,1903,26697,89921,89921,26697,1903,7,7,3824,
%U A168292 85660,463520,796594,463520,85660,3824,7,7,7665,268530,2224350,6068400
%N A168292 T(n,k) = 24*A046802(n+1,k+1) - 9*A008518(n,k) - 8*A007318(n,k), triangle read by rows (0 <= k <= n).
%F A168292 E.g.f.: 24*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 9*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - 8*exp(t*(1 + x)).
%e A168292 Triangle begins:
%e A168292      7;
%e A168292      7,    7;
%e A168292      7,   38,     7;
%e A168292      7,   99,    99,      7;
%e A168292      7,  220,   546,    220,      7;
%e A168292      7,  461,  2236,   2236,    461,      7;
%e A168292      7,  942,  8001,  15596,   8001,    942,     7;
%e A168292      7, 1903, 26697,  89921,  89921,  26697,  1903,    7;
%e A168292      7, 3824, 85660, 463520, 796594, 463520, 85660, 3824, 7;
%e A168292       ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%o A168292 (Maxima)
%o A168292 A123125(n, k) := sum((-1)^(k - j)*(binomial(n - j, k - j))*stirling2(n, j)*j!, j, 0, k)$
%o A168292 A046802(n, k) := sum(binomial(n - 1, r)*A123125(r, k - 1), r, k - 1, n - 1)$
%o A168292 A008518(n, k) := A123125(n, k) + A123125(n, k + 1)$
%o A168292 T(n, k) := 24*A046802(n + 1, k + 1) - 9*A008518(n, k) - 8*binomial(n, k)$
%o A168292 create_list(T(n, k), n, 0, 10, k, 0, n);
%o A168292 /* _Franck Maminirina Ramaharo_, Oct 21 2018 */
%Y A168292 Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y A168292 Cf. A142147, A142175, A168287, A168288, A168289, A168290, A168291, A168293.
%K A168292 nonn,easy,less,tabl
%O A168292 0,1
%A A168292 _Roger L. Bagula_, Nov 22 2009
%E A168292 Edited, new name from _Franck Maminirina Ramaharo_, Oct 21 2018