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A168287 T(n,k) = 2*A046802(n+1,k+1) - A007318(n,k), triangle read by rows (0 <= k <= n).

%I A168287 #12 Oct 22 2018 10:34:42
%S A168287 1,1,1,1,4,1,1,11,11,1,1,26,60,26,1,1,57,252,252,57,1,1,120,931,1746,
%T A168287 931,120,1,1,247,3201,10187,10187,3201,247,1,1,502,10534,53542,89788,
%U A168287 53542,10534,502,1,1,1013,33698,262466,688976,688976,262466,33698,1013
%N A168287 T(n,k) = 2*A046802(n+1,k+1) - A007318(n,k), triangle read by rows (0 <= k <= n).
%F A168287 E.g.f.: 2*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - exp(t*(1 + x)).
%e A168287 Triangle begins:
%e A168287      1;
%e A168287      1,    1;
%e A168287      1,    4,     1;
%e A168287      1,   11,    11,     1;
%e A168287      1,   26,    60,    26,     1;
%e A168287      1,   57,   252,   252,    57,     1;
%e A168287      1,  120,   931,  1746,   931,   120,     1;
%e A168287      1,  247,  3201, 10187, 10187,  3201,   247,   1;
%e A168287      1,  502, 10534, 53542, 89788, 53542, 10534, 502, 1;
%e A168287      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%t A168287 p[t_] = 2*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - Exp[t*(1 + x)];
%t A168287 Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten
%o A168287 (Maxima)
%o A168287 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 A168287 T(n, k) := 2*A046802(n + 1, k + 1) - binomial(n, k)$
%o A168287 create_list(T(n, k), n, 0, 10, k, 0, n);
%o A168287 /* _Franck Maminirina Ramaharo_, Oct 21 2018 */
%Y A168287 Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y A168287 Cf. A142147, A142175, A168288, A168289, A168290, A168291, A168292, A168293.
%K A168287 nonn,tabl,easy
%O A168287 0,5
%A A168287 _Roger L. Bagula_, Nov 22 2009
%E A168287 Edited, and new name by _Franck Maminirina Ramaharo_, Oct 21 2018