A157963 -id:A157963 - cp's OEIS frontend

A158474 Triangle read by rows generated from (x-1)(x-2)(x-4)*...

1, 1, -1, 1, -3, 2, 1, -7, 14, -8, 1, -15, 70, -120, 64, 1, -31, 310, -1240, 1984, -1024, 1, -63, 1302, -11160, 41664, -64512, 32768, 1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152, 1, -255, 21590, -777240, 12850368, -99486720, 353730560

Offset: 0

Gary W. Adamson, Mar 20 2009

Row sum of the unsigned triangle = A028361: (1, 2, 6, 30, 270, 4590, ...).
Right border of the unsigned triangle = A006125: (1, 1, 2, 8, 64, 1024, ...).
From Philippe Deléham, Mar 20 2009: (Start)
Unsigned triangle: A077957(n) DELTA A007179(n+1) = [1,0,2,0,4,0,8,0,16,0,32,0,...]DELTA[1,1,4,6,16,28,64,120,256,496,...], where DELTA is the operator defined in A084938.
Signed triangle: [1,0,2,0,4,0,8,0,16,0,...]DELTA[-1,-1,-4,-6,-16,-28,-64,...]. (End)

			First few rows of the triangle =
1;
1,  -1;
1,  -3,     2;
1,  -7,    14,     -8;
1, -15,    70,   -120,       64;
1, -31,   310,  -1240,     1984,    -1024;
1, -63,  1302, -11160,    41664,   -64512,     32768;
1,-127,  5334, -94488,   755904, -2731008,   4161536,  -2097152;
1,-255, 21590,-777240, 12850368,-99486720, 353730560,-534773760, 268435456;
...
Example: row 3 = x^3 - 7x^2 + 14x - 8 = (x-1)*(x-2)*(x-4).

Cf. A028361, A006125.

Cf. A157963, A135950. - R. J. Mathar, Mar 20 2009

A158474 := proc(n,k) mul(x-2^j,j=0..n-1) ; expand(%); coeftayl(%,x=0,n-k) ; end proc: # R. J. Mathar, Aug 27 2011

{{1}}~Join~Table[Reverse@ CoefficientList[Fold[#1 (x - #2) &, 1, 2^Range[0, n]], x], {n, 0, 7}] // Flatten (* Michael De Vlieger, Dec 22 2016 *)

T(n,k) = coefficient [x^(n-k)] of (x-1)*(x-2)*(x-4)*...*(x-2^(n-1)).
T(n,k) = (-1)^k*(Sum_{j=0..k} T(k,j)*2^((k-j)*n))/(Product_{i=1..k} (2^i-1)) for n >= 0 and k > 0, i.e., e.g.f. of col k > 0 is: (-1)^k*(Sum_{j=0..k} T(k,j)* exp(2^(k-j)*t))/(Product_{i=1..k} (2^i-1)). - Werner Schulte, Dec 18 2016
T(n,k)/T(k,k) = A022166(n,k) for 0 <= k <= n. - Werner Schulte, Dec 21 2016

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

Original entry on oeis.org

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

Roger L. Bagula, Mar 06 2009

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

			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;

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

Cf. this sequence (q=-2), A158020 (q=-1), A007318 (q=1), A157963 (q=2).
Cf. A135950 (q=2; alternative).
Cf. A022166, A135950.

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 *)

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)

Edited by G. C. Greubel, Nov 29 2021

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A158474 Triangle read by rows generated from (x-1)(x-2)(x-4)*...

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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.

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cp's OEIS Frontend

A158474 Triangle read by rows generated from (x-1)*(x-2)*(x-4)*...

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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.

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Keywords

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Examples

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Crossrefs

Programs

Mathematica

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A158474 Triangle read by rows generated from (x-1)(x-2)(x-4)*...