%I #12 Jan 04 2022 02:26:11
%S 1,-1,1,1,0,-2,1,-1,-1,1,2,1,-3,1,1,2,1,-1,-4,-2,0,3,3,-4,1,-1,-3,-4,
%T -3,1,5,8,5,-1,-4,-5,-3,3,6,-5,1,1,4,8,11,10,5,-5,-15,-19,-17,-7,5,13,
%U 9,7,-1,-7,-8,1,10,-6,1,-1,-5,-13,-24,-34,-39,-34,-17,9,38,59,63,50,26,-6,-28,-36,-25,-9,2,13,17,8,-5,-14,-4,15,-7,1
%N Irregular table with the coefficient [x^k] of Product_{j=1..n} (x^j - (1-x^j)/(1-x)) in row n, column 0<=k.
%H G. C. Greubel, <a href="/A156281/b156281.txt">Rows n = 0..25 of the irregular triangle, flattened</a>
%H L. Carlitz, <a href="http://dx.doi.org/10.1215/S0012-7094-48-01588-9">q-Bernoulli numbers and polynomials</a>, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.
%H L. Carlitz, J. Riordan, <a href="http://dx.doi.org/10.1215/S0012-7094-64-03136-9">Two element lattice permutation numbers and their q-generalization</a>, Duke Math. J. Volume 31, Number 3 (1964), 371-388.
%H John Shareshian and Michelle L. Wachs, <a href="http://arxiv.org/abs/math/0608274">q-Eulerian Polynomials: Excedance Number and Major Index</a>, arXiv:math/0608274 [math.CO], 2006, page 3.
%F T(n, k) = [x^k]( Product_{j=1..n} (x^j - (1-x^j)/(1-x)) ).
%F Sum_{k=0..binomial(n+1,2)} T(n, k) = A000007(n).
%e Irregular triangle begins as:
%e 1;
%e -1, 1;
%e 1, 0, -2, 1;
%e -1, -1, 1, 2, 1, -3, 1;
%e 1, 2, 1, -1, -4, -2, 0, 3, 3, -4, 1;
%e -1, -3, -4, -3, 1, 5, 8, 5, -1, -4, -5, -3, 3, 6, -5, 1;
%t T[n_]:= CoefficientList[(-1)^n*(1-x)^(-n)*Product[1 -2*x^j +x^(j+1), {j, n}], x];
%t Table[T[n], {n, 0, 10}] //Flatten (* modified by _G. C. Greubel_, Jan 03 2022 *)
%o (Sage)
%o def T(n,k): return ( (-1)^n*(1-x)^(-n)*product( 1 -2*x^j +x^(j+1) for j in (1..n)) ).series(x, 1+binomial(n+1,2)).list()[k]
%o flatten([[T(n,k) for k in (0..binomial(n+1,2))] for n in (0..10)]) # _G. C. Greubel_, Jan 03 2022
%K tabf,sign
%O 0,6
%A _Roger L. Bagula_, Feb 07 2009
%E Edited by _G. C. Greubel_, Jan 03 2022
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