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A231599
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T(n,k) is the coefficient of x^k in Product_{i=1..n} (1-x^i); triangle T(n,k), n >= 0, 0 <= k <= A000217(n), read by rows.
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15
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1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 0, 2, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 1, 1, -1, -1, -1, 0, 0, 1, 1, -1, 1, -1, -1, 0, 0, 1, 0, 2, 0, -1, -1, -1, -1, 0, 2, 0, 1, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 0, 1, 1, 0, -1, -1, -2, 0
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OFFSET
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0,20
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COMMENTS
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The sum of each row is 0. The even rows are symmetric; in the odd rows numbers with the same absolute value and opposed signum are symmetric to each other.
The odd rows where n mod 4 = 3 have the central value 0.
The even rows where n mod 4 = 0 have positive central values. They form the sequence A269298 and are also the rows maximal values.
A086376 contains the maximal values of each row, A160089 the maximal absolute values, and A086394 the absolute parts of the minimal values.
Rows of this triangle can be used to efficiently calculate values of A026807.
(End)
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LINKS
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FORMULA
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T(n,k) = [x^k] Product_{i=1..n} (1-x^i).
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EXAMPLE
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For n=2 the corresponding polynomial is (1-x)*(1-x^2) = 1 -x - x^2 + x^3.
Irregular triangle starts:
k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
n
0 1
1 1 -1
2 1 -1 -1 1
3 1 -1 -1 0 1 1 -1
4 1 -1 -1 0 0 2 0 0 -1 -1 1
5 1 -1 -1 0 0 1 1 1 -1 -1 -1 0 0 1 1 -1
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MAPLE
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T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))
(expand(mul(1-x^i, i=1..n))):
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MATHEMATICA
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Table[If[k == 0, 1, Coefficient[Product[(1 - x^i), {i, n}], x^k]], {n, 0, 6}, {k, 0, (n^2 + n)/2}] // Flatten (* Michael De Vlieger, Mar 04 2018 *)
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PROG
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(PARI) row(n) = pol = prod(i=1, n, 1 - x^i); for (i=0, poldegree(pol), print1(polcoeff(pol, i), ", ")); \\ Michel Marcus, Dec 21 2013
(Python)
from sympy import poly, symbols
def a231599_row(n):
if n == 0:
return [1]
x = symbols('x')
p = 1
for i in range(1, n+1):
p *= poly(1-x**i)
p = p.all_coeffs()
return p[::-1]
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CROSSREFS
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Cf. A269298 (central nonzero values).
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KEYWORD
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AUTHOR
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STATUS
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approved
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