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T(n,k) = 3*A046802(n+1,k+1) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).
8

%I #9 Oct 22 2018 10:34:58

%S 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 1389,177,1,1,367,4791,15263,15263,4791,367,1,1,749,15787,80285,

%U 134647,80285,15787,749,1,1,1515,50529,393657,1033401,1033401,393657,50529

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

%F E.g.f.: 3*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*exp(t*(1 + x)).

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 5, 1;

%e 1, 15, 15, 1;

%e 1, 37, 87, 37, 1;

%e 1, 83, 373, 373, 83, 1;

%e 1, 177, 1389, 2609, 1389, 177, 1;

%e 1, 367, 4791, 15263, 15263, 4791, 367, 1;

%e 1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1;

%e ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

%t p[t_] = 3*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 2*Exp[t*(1 + x)];

%t Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

%o (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 T(n, k) := 3*A046802(n + 1, k + 1) - 2*binomial(n, k)$

%o create_list(T(n, k), n, 0, 10, k, 0, n);

%o /* _Franck Maminirina Ramaharo_, Oct 21 2018 */

%Y Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

%Y Cf. A142147, A142175, A168287, A168289, A168290, A168291, A168292, A168293.

%K nonn,easy,tabl

%O 0,5

%A _Roger L. Bagula_, Nov 22 2009

%E Edited, and new name by _Franck Maminirina Ramaharo_, Oct 21 2018