cp's OEIS Frontend

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.

%I A340555 #7 Feb 03 2021 17:46:11
%S A340555 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 A340555 10725,0,4095,0,525,0,15,0,0,31,0,4805,0,195083,0,3260673,0,27172275,
%U A340555 0,124992465,0,336518175,0,548043885,0,548043885,0,336518175,0,124992465,0,27172275,0,3260673,0,195083,0,4805,0,31,0
%N 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.
%F A340555 A340555(n, k) = -A340263(n, k) * (k mod 2).
%e A340555 Triangle begins:
%e A340555 [0] [0]
%e A340555 [1] [0, 1, 0]
%e A340555 [2] [0, 3, 0, 3, 0]
%e A340555 [3] [0, 7, 0, 49, 0, 49, 0, 7, 0]
%e A340555 [4] [0, 15, 0, 525, 0, 4095, 0, 10725, 0, 10725, 0, 4095, 0, 525, 0, 15, 0]
%e A340555 [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 A340555 CoeffList := p -> [op(PolynomialTools:-CoefficientList(p, x)),0]:
%p A340555 Tpoly := proc(n) (2^n-1)*2^(-n-1)*((x+1)^(2^n) - (x-1)^(2^n)) end:
%p A340555 seq(print(CoeffList(Tpoly(n))), n=0..5);
%o A340555 (SageMath)
%o A340555 def A340555():
%o A340555     a, b, c = 1, 1, 1
%o A340555     yield [0]
%o A340555     while True:
%o A340555         c *= 2
%o A340555         a *= b
%o A340555         b = sum(binomial(c, 2 * k) * x ^ (2 * k) for k in range(c + 1))
%o A340555         q = ((b - (c - 1) * x * a)).list()
%o A340555         yield [-q[i] * (i % 2) for i in range(c + 1)]
%o A340555 A340555_row = A340555()
%o A340555 for _ in range(6):
%o A340555     print(next(A340555_row))
%Y A340555 Cf. A340263.
%K A340555 nonn,tabf
%O A340555 0,6
%A A340555 _Peter Luschny_, Jan 11 2021