A340555 - OEIS

%I #7 Feb 03 2021 17:46:11

%S 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,

%T 10725,0,4095,0,525,0,15,0,0,31,0,4805,0,195083,0,3260673,0,27172275,

%U 0,124992465,0,336518175,0,548043885,0,548043885,0,336518175,0,124992465,0,27172275,0,3260673,0,195083,0,4805,0,31,0

%N 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.

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

%e Triangle begins:

%e [0] [0]

%e [1] [0, 1, 0]

%e [2] [0, 3, 0, 3, 0]

%e [3] [0, 7, 0, 49, 0, 49, 0, 7, 0]

%e [4] [0, 15, 0, 525, 0, 4095, 0, 10725, 0, 10725, 0, 4095, 0, 525, 0, 15, 0]

%e [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]

%p CoeffList := p -> [op(PolynomialTools:-CoefficientList(p, x)),0]:

%p Tpoly := proc(n) (2^n-1)*2^(-n-1)*((x+1)^(2^n) - (x-1)^(2^n)) end:

%p seq(print(CoeffList(Tpoly(n))), n=0..5);

%o (SageMath)

%o def A340555():

%o     a, b, c = 1, 1, 1

%o     yield [0]

%o     while True:

%o         c *= 2

%o         a *= b

%o         b = sum(binomial(c, 2 * k) * x ^ (2 * k) for k in range(c + 1))

%o         q = ((b - (c - 1) * x * a)).list()

%o         yield [-q[i] * (i % 2) for i in range(c + 1)]

%o A340555_row = A340555()

%o for _ in range(6):

%o     print(next(A340555_row))

%Y Cf. A340263.

%K nonn,tabf

%O 0,6

%A _Peter Luschny_, Jan 11 2021