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A157785 Triangle of coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.

%I A157785 #15 Nov 30 2021 05:36:03
%S A157785 1,1,-1,-2,1,1,-8,6,3,-1,64,-40,-30,5,1,1024,-704,-440,110,11,-1,
%T A157785 -32768,21504,14784,-3080,-462,21,1,-2097152,1409024,924672,-211904,
%U A157785 -26488,1806,43,-1,268435456,-178257920,-119767040,26199040,3602368,-204680,-7310,85,1
%N A157785 Triangle of coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.
%C A157785 Triangle T(n,k), 0 <= k <= n, read by rows given by [1, q-1, q^2, q^3-q, q^4, q^5-q^2, q^6, q^7-q^3, q^8, ...] DELTA [-1, 0, -q, 0, -q^2, 0, -q^3, 0, -q^4, 0, ...] (for q=-2) = [1, -3, 4, -6, 16, -36, 64,...] DELTA [ -1, 0, 2, 0, -4, 0, 8, 0, -16, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 10 2009
%H A157785 G. C. Greubel, <a href="/A157785/b157785.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157785 Sum_{k=0..n} T(n, k) = 0^n.
%F A157785 From _G. C. Greubel_, Nov 29 2021: (Start)
%F A157785 T(n, k) = [x^k] coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.
%F A157785 T(n, k) = [x^k] Product_{j=0..n-1} (q^j - x). (End)
%e A157785 Triangle begins as:
%e A157785           1;
%e A157785           1,         -1;
%e A157785          -2,          1,          1;
%e A157785          -8,          6,          3,       -1;
%e A157785          64,        -40,        -30,        5,       1;
%e A157785        1024,       -704,       -440,      110,      11,      -1;
%e A157785      -32768,      21504,      14784,    -3080,    -462,      21,     1;
%e A157785    -2097152,    1409024,     924672,  -211904,  -26488,    1806,    43, -1;
%e A157785   268435456, -178257920, -119767040, 26199040, 3602368, -204680, -7310, 85, 1;
%t A157785 p[x_, n_, q_]:= q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
%t A157785 Table[CoefficientList[Series[p[x, n, -2], {x,0,n}], x], {n,0,10}]//Flatten (* _G. C. Greubel_, Nov 29 2021 *)
%Y A157785 Cf. this sequence (q=-2), A158020 (q=-1), A007318 (q=1), A157963 (q=2).
%Y A157785 Cf. A135950 (q=2; alternative).
%Y A157785 Cf. A022166, A135950.
%K A157785 sign,tabl
%O A157785 0,4
%A A157785 _Roger L. Bagula_, Mar 06 2009
%E A157785 Edited by _G. C. Greubel_, Nov 29 2021