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%I #15 Oct 22 2018 10:35:07
%S 1,1,1,1,8,1,1,36,36,1,1,133,420,133,1,1,449,3334,3334,449,1,1,1446,
%T 21939,49364,21939,1446,1,1,4534,130044,560957,560957,130044,4534,1,1,
%U 13991,724222,5459561,10284514,5459561,724222,13991,1,1,42747,3880014
%N T(n,k) = (1/4)*A007318(n,k) - (3/2)*A008292(n+1,k+1) + (9/4)*A060187(n+1,k+1), triangle read by rows (0 <= k <= n).
%C Row n gives the coefficients in the expansion of (1/4)*(1 + x)^n + (9/4)*2^n*(1 - x)^(1 + n)*Phi(x, -n, 1/2) - (3/2)*(1 - x)^(n + 2)*Phi(x, -1 - n, 1), where Phi is the Lerch transcendant.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lerch_zeta_function">Lerch zeta function</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>
%F E.g.f.: (exp((1 + x)*y) - 6*(1 - x)^2*exp(y*(1 - x))/(1 - x*exp(y*(1 - x)))^2 + 9*(1 - x)*exp((1 - x)*y)/(1 - x*exp(2*(1 - x)*y)))/4. - _Franck Maminirina Ramaharo_, Oct 20 2018
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 8, 1;
%e 1, 36, 36, 1;
%e 1, 133, 420, 133, 1;
%e 1, 449, 3334, 3334, 449, 1;
%e 1, 1446, 21939, 49364, 21939, 1446, 1;
%e 1, 4534, 130044, 560957, 560957, 130044, 4534, 1;
%e ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%t p[x_, n_] = 1/4*(1 + x)^n + 9/4*2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2] - 3/2*(1 - x)^(2 + n)*PolyLog[-1 - n, x]/x;
%t Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 0, 10}]// Flatten
%o (Maxima)
%o A008292(n, k) := sum((-1)^j*(k - j)^n*binomial(n + 1, j), j, 0, k)$
%o A060187(n, k) := sum((-1)^(k - j)*binomial(n, k - j)*(2*j - 1)^(n - 1), j, 1, k)$
%o T(n, k) := (binomial(n, k) - 6*A008292(n + 1, k + 1) + 9*A060187(n + 1, k + 1))/4$
%o create_list(T(n, k), n, 0, 10, k, 0, n);
%o /* _Franck Maminirina Ramaharo_, Oct 20 2018 */
%Y Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y Cf. A142147, A168287, A168288, A168289, A168290, A168291, A168292, A168293.
%K nonn,easy,tabl
%O 0,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 16 2008
%E Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 19 2018
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