cp's OEIS Frontend

A373308 Expansion of Product_{n>=0} (1 - x^(2^n))^3.

%I A373308 #18 Jun 25 2024 19:20:14
%S A373308 1,-3,0,8,-9,3,8,-24,15,19,-24,0,17,-27,0,64,-57,-21,64,-72,33,53,-72,
%T A373308 24,17,-51,24,64,-81,27,64,-192,135,107,-192,120,1,-171,120,152,-183,
%U A373308 -27,152,-192,87,163,-192,0,89,-75,0,136,-129,-21,136,-216,111,179,-216,0,145,-219,0,512
%N A373308 Expansion of Product_{n>=0} (1 - x^(2^n))^3.
%C A373308 Equals the self-convolution cube of the signed Thue-Morse sequence A106400: [1, -1, -1, 1, -1, 1, 1, -1, ...].
%C A373308 Conjecture: a(3*n) == A001285(n) (mod 3) for n >= 0, where A001285 is the Thue-Morse sequence: [1, 2, 2, 1, 2, 1, 1, 2, ...].
%C A373308 Conjecture: a(3*n+1) and a(3*n+2) are divisible by 3 for n >= 0.
%C A373308 Conjecture: a(n) = 0 iff n = 3*A000695(k) - 1 for k >= 1, where A000695 lists sums of distinct powers of 4.
%H A373308 Paul D. Hanna, <a href="/A373308/b373308.txt">Table of n, a(n) for n = 0..16400</a>
%F A373308 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A373308 (1) A(x) = Product_{n>=0} (1 - x^(2^n))^3.
%F A373308 (2) A(x) = (1-x)^3 * A(x^2).
%F A373308 (3) a(n) = Sum_{k=0..n} A106400(k) * A106407(n-k+1) for n >= 0.
%e A373308 G.f.: A(x) = 1 - 3*x + 8*x^3 - 9*x^4 + 3*x^5 + 8*x^6 - 24*x^7 + 15*x^8 + 19*x^9 - 24*x^10 + 17*x^12 - 27*x^13 + 64*x^15 - 57*x^16 - 21*x^17 + 64*x^18 + ...
%e A373308 where A(x) = (1-x)^3 * (1-x^2)^3 * (1-x^4)^3 * (1-x^8)^3 * (1-x^16)^3 * ...
%e A373308 Notice that a(n) = 0 at n = [2, 11, 14, 47, 50, 59, 62, 191, 194, 203, 206, 239, 242, 251, 254, 767, ...] which appears to equal 3*A000695(k) - 1 for k >= 1.
%o A373308 (PARI) {a(n) = my(A = prod(k=0,#binary(n), (1 - x^(2^k))^3 +x*O(x^n))); polcoeff(A, n)}
%o A373308 for(n=0,60, print1(a(n),", "))
%Y A373308 Cf. A373309, A106400, A106407, A001285, A000695.
%K A373308 sign,look
%O A373308 0,2
%A A373308 _Paul D. Hanna_, Jun 20 2024