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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
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified April 24 00:30 EDT 2024. Contains 371917 sequences. (Running on oeis4.)