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A157963
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,-q^5,0,...] (for q=2) = [1,1,4,6,16,28,64,...] DELTA [ -1,0,-2,0,-4,0,-8,0,-16,0,...] where DELTA is the operator defined in A084938.
3
1, 1, -1, 2, -3, 1, 8, -14, 7, -1, 64, -120, 70, -15, 1, 1024, -1984, 1240, -310, 31, -1, 32768, -64512, 41664, -11160, 1302, -63, 1, 2097152, -4161536, 2731008, -755904, 94488, -5334, 127, -1, 268435456, -534773760, 353730560, -99486720, 12850368
OFFSET
0,4
COMMENTS
Row sums equal 0^n.
Row n contains the coefficients of Product_{j=0..n-1} (2^j*x-1), highest coefficient first. - Alois P. Heinz, Mar 26 2012
The elements of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^k*A022166(n,k). - R. J. Mathar, Mar 26 2013
LINKS
Alois P. Heinz, Rows n = 0..44
FORMULA
T(n,k) = (-1)^n*A135950(n,k). T(n,0) = A006125(n).
T(n,k) = [x^(n-k)] Product_{j=0..n-1} (2^j*x-1). - Alois P. Heinz, Mar 26 2012
EXAMPLE
Triangle begins :
1;
1, -1;
2, -3, 1;
8, -14, 7, -1;
64, -120, 70, -15, 1;
MAPLE
T:= n-> seq (coeff (mul (2^j*x-1, j=0..n-1), x, n-k), k=0..n):
seq (T(n), n=0..10); # Alois P. Heinz, Mar 26 2012
MATHEMATICA
row[n_] := CoefficientList[(-1)^n QPochhammer[x, 2, n] + O[x]^(n+1), x] // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 26 2016 *)
KEYWORD
sign,tabl
AUTHOR
Philippe Deléham, Mar 10 2009
STATUS
approved