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 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. 5

%I #30 Apr 09 2021 10:37:12

%S 1,1,-1,1,1,-3,6,-3,1,1,-7,28,-49,70,-49,28,-7,1,1,-15,120,-525,1820,

%T -4095,8008,-10725,12870,-10725,8008,-4095,1820,-525,120,-15,1

%N 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.

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

%H Peter Luschny, <a href="/A340263/b340263.txt">Table of n, a(n) for n = 0..1031</a>

%F 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).

%e Polynomials begin:

%e [0] 1;

%e [1] x^2 - x + 1;

%e [2] x^4 - 3*x^3 + 6*x^2 - 3*x + 1;

%e [3] x^8 - 7*x^7 + 28*x^6 - 49*x^5 + 70*x^4 - 49*x^3 + 28*x^2 - 7*x + 1;

%e Triangle begins:

%e [0] [1]

%e [1] [1, -1, 1]

%e [2] [1, -3, 6, -3, 1]

%e [3] [1, -7, 28, -49, 70, -49, 28, -7, 1]

%e [4] [1, -15, 120, -525, 1820, -4095, 8008, -10725, 12870, -10725, 8008, -4095, 1820, -525, 120, -15, 1]

%p A340263_row := proc(n) local a, b;

%p if n = 0 then return [1] fi;

%p b := n -> add(binomial(2^n, 2*k)*x^(2*k), k = 0..2^n);

%p a := n -> x*mul(b(k), k = 0..n);

%p expand(b(n) - (2^n-1)*a(n-1));

%p [seq(coeff(%, x, j), j = 0..2^n)] end:

%p for n from 0 to 5 do A340263_row(n) od;

%p # Alternatively:

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

%p Tpoly := n -> ((1-x)^(2^n) + 2^(-n)*((2^n-1)*(x-1)^(2^n) + (x + 1)^(2^n)))/2:

%p seq(print(CoeffList(Tpoly(n))), n=0..5); # _Peter Luschny_, Feb 03 2021

%o (SageMath)

%o def A340263():

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

%o yield [1]

%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 yield ((b - (c - 1) * x * a)).list()

%o A340263_row = A340263()

%o for _ in range(6):

%o print(next(A340263_row))

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

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

%Y Cf. A340312.

%K sign,tabf,look

%O 0,6

%A _Peter Luschny_, Jan 06 2021

%E Shorter name by _Peter Luschny_, Feb 03 2021

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Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)