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%I #12 Sep 08 2022 08:45:51
%S 1,1,1,1,-8,1,1,-69,-69,1,1,-374,-974,-374,1,1,-1735,-9330,-9330,
%T -1735,1,1,-7496,-74969,-152144,-74969,-7496,1,1,-31241,-545083,
%U -1983485,-1983485,-545083,-31241,1,1,-127754,-3724784,-22499414,-39828194,-22499414,-3724784,-127754,1
%N Triangle read by rows: T(n, m, q) = (1-q^n)*Eulerian(n+1, m) - (1-q^n) + 1, with q = 2.
%C Row sums are: {1, 2, -6, -136, -1720, -22128, -317072, -5119616, -92532096, -1854311680, -40834875136, ...}.
%H G. C. Greubel, <a href="/A174728/b174728.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, m, q) = (1 - q^n)*Eulerian(n + 1, m) - (1 - q^n) + 1, where q = 2.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -8, 1;
%e 1, -69, -69, 1;
%e 1, -374, -974, -374, 1;
%e 1, -1735, -9330, -9330, -1735, 1;
%e 1, -7496, -74969, -152144, -74969, -7496, 1;
%e 1, -31241, -545083, -1983485, -1983485, -545083, -31241, 1;
%t Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j,0,k+1}];
%t With[{q = 2}, Table[(1-q^n)*(Eulerian[n+1, m]-1)+1, {n,0,10}, {m,0, n}] ]//Flatten (* _G. C. Greubel_, Apr 20 2019 *)
%o (PARI) q=2; {eulerian(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n)};
%o for(n=0,10, for(k=0,n, print1((1-q^n)*(eulerian(n+1,k)-1)+1, ", "))) \\ _G. C. Greubel_, Apr 20 2019
%o (Magma) q:=2; Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >; [[(1-q^n)*(Eulerian(n+1,k)-1) +1: k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Apr 20 2019
%o (Sage)
%o q=2;
%o def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
%o [[(1-q^n)*(Eulerian(n+1,k)-1)+1 for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 20 2019
%K sign,tabl
%O 0,5
%A _Roger L. Bagula_, Mar 28 2010
%E Edited by _G. C. Greubel_, Apr 20 2019
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