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A142175 Triangle read by rows: T(n,k) = (1/4)*(A007318(n,k) - 6*A008292(n+1,k+1) + 9*A060187(n+1,k+1)).

%I A142175 #23 Dec 31 2024 16:59:24
%S A142175 1,1,1,1,8,1,1,36,36,1,1,133,420,133,1,1,449,3334,3334,449,1,1,1446,
%T A142175 21939,49364,21939,1446,1,1,4534,130044,560957,560957,130044,4534,1,1,
%U A142175 13991,724222,5459561,10284514,5459561,724222,13991,1,1,42747,3880014,48160170,154214412,154214412,48160170,3880014,42747,1
%N A142175 Triangle read by rows: T(n,k) = (1/4)*(A007318(n,k) - 6*A008292(n+1,k+1) + 9*A060187(n+1,k+1)).
%C A142175 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 A142175 G. C. Greubel, <a href="/A142175/b142175.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A142175 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lerch_zeta_function">Lerch zeta function</a>
%H A142175 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>
%F A142175 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 A142175 Triangle begins:
%e A142175      1;
%e A142175      1,    1;
%e A142175      1,    8,      1;
%e A142175      1,   36,     36,      1;
%e A142175      1,  133,    420,    133,      1;
%e A142175      1,  449,   3334,   3334,    449,      1;
%e A142175      1, 1446,  21939,  49364,  21939,   1446,    1;
%e A142175      1, 4534, 130044, 560957, 560957, 130044, 4534, 1;
%e A142175       ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%t A142175 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 A142175 Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 0, 10}]// Flatten
%o A142175 (Maxima)
%o A142175 A008292(n, k) := sum((-1)^j*(k - j)^n*binomial(n + 1, j), j, 0, k)$
%o A142175 A060187(n, k) := sum((-1)^(k - j)*binomial(n, k - j)*(2*j - 1)^(n - 1), j, 1, k)$
%o A142175 T(n, k) := (binomial(n, k) - 6*A008292(n + 1, k + 1) + 9*A060187(n + 1, k + 1))/4$
%o A142175 create_list(T(n, k), n, 0, 10, k, 0, n);
%o A142175 /* _Franck Maminirina Ramaharo_, Oct 20 2018 */
%o A142175 (Magma)
%o A142175 A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
%o A142175 A142175:= func< n,k | (Binomial(n,k) - 6*EulerianNumber(n+1,k) + 9*A060187(n+1,k+1))/4 >;
%o A142175 [A142175(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 30 2024
%o A142175 (SageMath)
%o A142175 # from sage.all import * # (use for Python)
%o A142175 from sage.combinat.combinat import eulerian_number
%o A142175 def A060187(n,k): return sum(pow(-1, k-j)*binomial(n, k-j)*pow(2*j-1, n-1) for j in range(1,k+1))
%o A142175 def A142175(n,k): return (binomial(n,k) - 6*eulerian_number(n+1,k) +9*A060187(n+1,k+1))//4
%o A142175 print(flatten([[A142175(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Dec 30 2024
%Y A142175 Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y A142175 Cf. A142147, A168287, A168288, A168289, A168290, A168291, A168292, A168293.
%K A142175 nonn,easy,tabl
%O A142175 0,5
%A A142175 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 16 2008
%E A142175 Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 19 2018