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A142075 Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.

%I A142075 #15 Sep 08 2022 08:45:35
%S A142075 1,1,2,1,8,4,1,22,44,8,1,52,264,208,16,1,114,1208,2416,912,32,1,240,
%T A142075 4764,19328,19056,3840,64,1,494,17172,124952,249904,137376,15808,128,
%U A142075 1,1004,58432,705872,2499040,2823488,934912,64256,256,1,2026,191360,3641536,20965664,41931328,29132288,6123520,259328,512
%N A142075 Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.
%C A142075 Same as A156365, except for the additional a(0) = 1 there.
%H A142075 G. C. Greubel, <a href="/A142075/b142075.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A142075 G.f.: 1/x/Q(0) -1/x, where Q(k) = 1 - x*(k+1)/( 1 - y*2*x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 17 2013
%F A142075 Sum_{k=1..n} T(n, k) = A000670(n), for n >= 1. - _G. C. Greubel_, Jun 07 2021
%e A142075 Triangle begins as:
%e A142075   1;
%e A142075   1,    2;
%e A142075   1,    8,     4;
%e A142075   1,   22,    44,      8;
%e A142075   1,   52,   264,    208,      16;
%e A142075   1,  114,  1208,   2416,     912,      32;
%e A142075   1,  240,  4764,  19328,   19056,    3840,     64;
%e A142075   1,  494, 17172, 124952,  249904,  137376,  15808,   128;
%e A142075   1, 1004, 58432, 705872, 2499040, 2823488, 934912, 64256, 256;
%e A142075   ...
%t A142075 (* First program *)
%t A142075 p[x_, n_]= (1-2*x)^(n+1)*PolyLog[-n, 2*x]/(2*x);
%t A142075 Table[CoefficientList[p[x, n], x], {n,12}]//Flatten
%t A142075 (* Second program *)
%t A142075 Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
%t A142075 Table[2^(k-1)*Eulerian[n, k-1], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 07 2021 *)
%o A142075 (Magma)
%o A142075 Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
%o A142075 [2^(k-1)*Eulerian(n,k-1): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jun 07 2021
%o A142075 (Sage)
%o A142075 def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
%o A142075 flatten([[2^(k-1)*Eulerian(n,k-1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jun 07 2021
%Y A142075 Essentially the same as A156365.
%Y A142075 Cf. A000670, A008292, A123125, A173018.
%K A142075 nonn,tabl
%O A142075 1,3
%A A142075 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008
%E A142075 Edited and new name by _Joerg Arndt_, Dec 30 2018