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A340555
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T(n, k) = [x^k] (2^n-1)*2^(-n-1)*((x+1)^(2^n) - (x-1)^(2^n)). Irregular triangle read by rows, for n >= 0 and 0 <= k <= 2^n.
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1
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0, 0, 1, 0, 0, 3, 0, 3, 0, 0, 7, 0, 49, 0, 49, 0, 7, 0, 0, 15, 0, 525, 0, 4095, 0, 10725, 0, 10725, 0, 4095, 0, 525, 0, 15, 0, 0, 31, 0, 4805, 0, 195083, 0, 3260673, 0, 27172275, 0, 124992465, 0, 336518175, 0, 548043885, 0, 548043885, 0, 336518175, 0, 124992465, 0, 27172275, 0, 3260673, 0, 195083, 0, 4805, 0, 31, 0
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refs;
listen;
history;
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internal format)
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OFFSET
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0,6
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
[0] [0]
[1] [0, 1, 0]
[2] [0, 3, 0, 3, 0]
[3] [0, 7, 0, 49, 0, 49, 0, 7, 0]
[4] [0, 15, 0, 525, 0, 4095, 0, 10725, 0, 10725, 0, 4095, 0, 525, 0, 15, 0]
[5] [0, 31, 0, 4805, 0, 195083, 0, 3260673, 0, 27172275, 0, 124992465, 0, 336518175, 0, 548043885, 0, 548043885, 0, 336518175, 0, 124992465, 0, 27172275, 0, 3260673, 0, 195083, 0, 4805, 0, 31, 0]
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MAPLE
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CoeffList := p -> [op(PolynomialTools:-CoefficientList(p, x)), 0]:
Tpoly := proc(n) (2^n-1)*2^(-n-1)*((x+1)^(2^n) - (x-1)^(2^n)) end:
seq(print(CoeffList(Tpoly(n))), n=0..5);
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PROG
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(SageMath)
a, b, c = 1, 1, 1
yield [0]
while True:
c *= 2
a *= b
b = sum(binomial(c, 2 * k) * x ^ (2 * k) for k in range(c + 1))
q = ((b - (c - 1) * x * a)).list()
yield [-q[i] * (i % 2) for i in range(c + 1)]
for _ in range(6):
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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