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A168290 T(n,k) = 5*A046802(n+1,k+1) - 4*A007318(n,k), triangle read by rows (0 <= k <= n).

%I A168290 #9 Oct 22 2018 10:34:08
%S A168290 1,1,1,1,7,1,1,23,23,1,1,59,141,59,1,1,135,615,615,135,1,1,291,2305,
%T A168290 4335,2305,291,1,1,607,7971,25415,25415,7971,607,1,1,1243,26293,
%U A168290 133771,224365,133771,26293,1243,1,1,2519,84191,656039,1722251,1722251,656039
%N A168290 T(n,k) = 5*A046802(n+1,k+1) - 4*A007318(n,k), triangle read by rows (0 <= k <= n).
%F A168290 E.g.f.: 5*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 4*exp(t*(1 + x)).
%e A168290 Triangle begins:
%e A168290     1;
%e A168290     1,    1;
%e A168290     1,    7,     1;
%e A168290     1,   23,    23,      1;
%e A168290     1,   59,   141,     59,      1;
%e A168290     1,  135,   615,    615,    135,      1;
%e A168290     1,  291,  2305,   4335,   2305,    291,     1;
%e A168290     1,  607,  7971,  25415,  25415,   7971,   607,    1;
%e A168290     1, 1243, 26293, 133771, 224365, 133771, 26293, 1243, 1;
%e A168290      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%t A168290 p[t_] = 5*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 4*Exp[t*(1 + x)];
%t A168290 Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[ t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten
%o A168290 (Maxima)
%o A168290 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 A168290 T(n, k) := 5*A046802(n + 1, k + 1) - 4*binomial(n, k)$
%o A168290 create_list(T(n, k), n, 0, 10, k, 0, n);
%o A168290 /* _Franck Maminirina Ramaharo_, Oct 21 2018 */
%Y A168290 Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y A168290 Cf. A142147, A142175, A168287, A168288, A168289, A168291, A168292, A168293.
%K A168290 nonn,tabl,easy
%O A168290 0,5
%A A168290 _Roger L. Bagula_, Nov 22 2009
%E A168290 Edited, new name by _Franck Maminirina Ramaharo_, Oct 21 2018