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