A340555 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.

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

Offset: 0

Author

Peter Luschny, Jan 11 2021

Keywords

• nonn
• tabf

Examples

			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]
		

Crossrefs

Cf. A340263.

Programs

Maple

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);

SageMath

def A340555():
    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)]
A340555_row = A340555()
for _ in range(6):
    print(next(A340555_row))

Formula

A340555(n, k) = -A340263(n, k) * (k mod 2).