A354272 - OEIS

%I #23 Aug 24 2022 10:12:27

%S 1,-1,0,1,1,0,-2,0,-1,0,-2,0,1,1,0,-4,0,2,0,-4,0,15,0,8,0,-36,0,8,0,

%T 15,0,-4,0,2,0,-4,0,1,1,0,-8,0,20,0,-24,0,58,0,-80,0,-92,0,120,0,147,

%U 0,384,0,-2108,0,880,0,3940,0,-3096,0,2288,0,-2136,0,-1803,0,-2136,0,2288,0,-3096,0,3940,0,880,0,-2108,0,384,0,147,0,120,0,-92,0,-80,0,58,0,-24,0,20,0,-8,0,1

%N Irregular triangle read by rows: coefficients of polynomials which are the product of all possible monic Littlewood polynomials of degree n.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Littlewood_polynomial">Littlewood polynomial</a>

%e The triangle T(n, k) begins

%e n\k  1 2  3 4  5 6  7 8 9 10 11 12  13 14 15 16 17 18 19 20 21 22 23 24 25

%e 0:   1

%e 1:  -1 0  1

%e 2:   1 0 -2 0 -1 0 -2 0 1

%e 3:   1 0 -4 0  2 0 -4 0 15 0  8  0 -36  0  8  0 15  0 -4  0  2  0 -4  0  1

%e ...

%e E.g., row 2: {1,0,-2,0,-1,0,-2,0,1} corresponds to polynomial 1-2x^2-x^4-2x^6+x^8.

%e Number of terms in each row equals A002064(n).

%o (Python)

%o from itertools import product

%o def mult_pol(s1, s2):

%o     res = [0]*(len(s1)+len(s2)-1)

%o     for o1,i1 in enumerate(s1):

%o         for o2,i2 in enumerate(s2):

%o             res[o1+o2] += i1*i2

%o     return res

%o out = []

%o for d in range(0, 5):

%o     startp = [1,]

%o     for i in product((1,-1),repeat = d):

%o         startp = mult_pol(startp, list(i)+[1,])

%o     out.extend(startp)

%o print(out)

%o (PARI) row(n) = { Vecrev(Vec(prod (k=2^n, 2^(n+1)-1, Pol(apply(d -> if (d, 1, -1), binary(k)))))) } \\ _Rémy Sigrist_, Jul 21 2022

%Y Cf. A020985, A002064 (row lengths).

%K sign,tabf

%O 0,7

%A _Gleb Ivanov_, May 22 2022