A106407 Expansion of x((1-x)(1-x^2)(1-x^4)(1-x^8)...)^2.
1, -2, -1, 4, -3, 2, 3, -8, 1, 6, -1, -4, 5, -6, -5, 16, -7, -2, 7, -12, 5, 2, -5, 8, 1, -10, -1, 12, -11, 10, 11, -32, 9, 14, -9, 4, 5, -14, -5, 24, -7, -10, 7, -4, -3, 10, 3, -16, 9, -2, -9, 20, -11, 2, 11, -24, 1, 22, -1, -20, 21, -22, -21, 64, -23, -18, 23, -28, 5, 18, -5, -8, 9, -10, -9, 28, -19, 10, 19, -48, 17, 14, -17, 20
Offset: 1
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
- Maciej Gawron, Piotr Miska, Maciej Ulas, Arithmetic properties of coefficients of power series expansion of Prod_{n>=0} (1-x^(2^n))^t, arXiv:1703.01955 [math.NT], 2017.
- Eric Rowland, A matrix generalization of a theorem of Fine, arXiv:1704.05872 [math.NT], 2017.
- Eric Rowland, A matrix generalization of a theorem of Fine, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A18.
Programs
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Mathematica
Table[Sum[(-1)^(DigitCount[k, 2, 1] + DigitCount[n-k-1, 2, 1]), {k, 0, n-1}], {n, 1, 80}] (* Vladimir Reshetnikov, Apr 13 2018 *) -
PARI
{a(n)=local(A,m); if(n<1,0,n--; A=1+x*O(x^n); m=1; while(m<=n, A*=(1-x^m); m*=2;); polcoeff(A^2,n))}
Formula
Euler transform of sequence b(n) where b(2^k)=-2 and zero otherwise.
G.f.: A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1^3*u6 + 6*u1^2*u2*u6 + 9*u1*u2^2*u6 - u3*u2^3.
G.f.: x(Product_{k>=0} (1-x^(2^k)))^2.
G.f.: A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=-v^3+4uvw+u^2w.
Comments