A340263 T(n, k) = [x^k] ((1-x)^(2^n) + 2^(-n)*((2^n-1)*(x-1)^(2^n) + (x+1)^(2^n)))/2. Irregular triangle read by rows, for n >= 0 and 0 <= k <= 2^n.

1, 1, -1, 1, 1, -3, 6, -3, 1, 1, -7, 28, -49, 70, -49, 28, -7, 1, 1, -15, 120, -525, 1820, -4095, 8008, -10725, 12870, -10725, 8008, -4095, 1820, -525, 120, -15, 1

Offset

0,6

Comments

Conjecture: for n >= 1 the polynomials are irreducible.

Links

Peter Luschny, Table of n, a(n) for n = 0..1031

Formula

Let p_n(x) = b(n) - (2^n-1)*a(n-1), b(n) = Sum_{k=0..2^n} binomial(2^n, 2*k)* x^(2*k), and a(n) = x*Product_{k=0..n} b(k). Then T(n, k) = [x^k] p_n(x).

Example

Polynomials begin:
[0] 1;
[1] x^2 - x + 1;
[2] x^4 - 3*x^3 + 6*x^2 - 3*x + 1;
[3] x^8 - 7*x^7 + 28*x^6 - 49*x^5 + 70*x^4 - 49*x^3 + 28*x^2 - 7*x + 1;
Triangle begins:
[0] [1]
[1] [1, -1, 1]
[2] [1, -3, 6, -3, 1]
[3] [1, -7, 28, -49, 70, -49, 28, -7, 1]
[4] [1, -15, 120, -525, 1820, -4095, 8008, -10725, 12870, -10725, 8008, -4095, 1820, -525, 120, -15, 1]

Maple

A340263_row := proc(n) local a, b;
if n = 0 then return [1] fi;
b := n -> add(binomial(2^n, 2*k)*x^(2*k), k = 0..2^n);
a := n -> x*mul(b(k), k = 0..n);
expand(b(n) - (2^n-1)*a(n-1));
[seq(coeff(%, x, j), j = 0..2^n)] end:
for n from 0 to 5 do A340263_row(n) od;
# Alternatively:
CoeffList := p -> [op(PolynomialTools:-CoefficientList(p, x))]:
Tpoly := n -> ((1-x)^(2^n) + 2^(-n)*((2^n-1)*(x-1)^(2^n) + (x + 1)^(2^n)))/2:
seq(print(CoeffList(Tpoly(n))), n=0..5); # Peter Luschny, Feb 03 2021

Prog

(SageMath)
def A340263():
    a, b, c = 1, 1, 1
    yield [1]
    while True:
        c *= 2
        a *= b
        b = sum(binomial(c, 2 * k) * x ^ (2 * k) for k in range(c + 1))
        yield ((b - (c - 1) * x * a)).list()
A340263_row = A340263()
for _ in range(6):
    print(next(A340263_row))

Crossrefs

Row sums are 2^(2^n - n - 1) = A016031(n-1).

Central terms of the rows are A037293(n) for n >= 2.

Cf. A340312.

Sequence in context: A216803 A094118 A200478 * A256158 A349237 A123060

Adjacent sequences: A340260 A340261 A340262 * A340264 A340265 A340266

Keyword

sign,tabf,look

Author

Peter Luschny, Jan 06 2021

Extensions

Shorter name by Peter Luschny, Feb 03 2021

Status

approved