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

1, 1, -1, -2, 1, 1, -8, 6, 3, -1, 64, -40, -30, 5, 1, 1024, -704, -440, 110, 11, -1, -32768, 21504, 14784, -3080, -462, 21, 1, -2097152, 1409024, 924672, -211904, -26488, 1806, 43, -1, 268435456, -178257920, -119767040, 26199040, 3602368, -204680, -7310, 85, 1

Offset: 0

Author

Roger L. Bagula, Mar 06 2009

Keywords

• sign
• tabl

Comments

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

Examples

			Triangle begins as:
          1;
          1,         -1;
         -2,          1,          1;
         -8,          6,          3,       -1;
         64,        -40,        -30,        5,       1;
       1024,       -704,       -440,      110,      11,      -1;
     -32768,      21504,      14784,    -3080,    -462,      21,     1;
   -2097152,    1409024,     924672,  -211904,  -26488,    1806,    43, -1;
  268435456, -178257920, -119767040, 26199040, 3602368, -204680, -7310, 85, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. this sequence (q=-2), A158020 (q=-1), A007318 (q=1), A157963 (q=2).

Cf. A135950 (q=2; alternative).

Cf. A022166, A135950.

Programs

Mathematica

p[x_, n_, q_]:= q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
Table[CoefficientList[Series[p[x, n, -2], {x,0,n}], x], {n,0,10}]//Flatten (* G. C. Greubel, Nov 29 2021 *)

Formula

Sum_{k=0..n} T(n, k) = 0^n.

From G. C. Greubel, Nov 29 2021: (Start)

T(n, k) = [x^k] coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.

T(n, k) = [x^k] Product_{j=0..n-1} (q^j - x). (End)

Extensions

Edited by G. C. Greubel, Nov 29 2021