A373309 -id:A373309 - cp's OEIS frontend

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

1, -3, 0, 8, -9, 3, 8, -24, 15, 19, -24, 0, 17, -27, 0, 64, -57, -21, 64, -72, 33, 53, -72, 24, 17, -51, 24, 64, -81, 27, 64, -192, 135, 107, -192, 120, 1, -171, 120, 152, -183, -27, 152, -192, 87, 163, -192, 0, 89, -75, 0, 136, -129, -21, 136, -216, 111, 179, -216, 0, 145, -219, 0, 512

Offset: 0

Paul D. Hanna, Jun 20 2024

Equals the self-convolution cube of the signed Thue-Morse sequence A106400: [1, -1, -1, 1, -1, 1, 1, -1, ...].
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, ...].
Conjecture: a(3*n+1) and a(3*n+2) are divisible by 3 for n >= 0.
Conjecture: a(n) = 0 iff n = 3*A000695(k) - 1 for k >= 1, where A000695 lists sums of distinct powers of 4.

			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 + ...
where A(x) = (1-x)^3 * (1-x^2)^3 * (1-x^4)^3 * (1-x^8)^3 * (1-x^16)^3 * ...
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.

Paul D. Hanna, Table of n, a(n) for n = 0..16400

Cf. A373309, A106400, A106407, A001285, A000695.

{a(n) = my(A = prod(k=0,#binary(n), (1 - x^(2^k))^3 +x*O(x^n))); polcoeff(A, n)}
for(n=0,60, print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Product_{n>=0} (1 - x^(2^n))^3.
(2) A(x) = (1-x)^3 * A(x^2).
(3) a(n) = Sum_{k=0..n} A106400(k) * A106407(n-k+1) for n >= 0.

A373313 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^3 ).

Original entry on oeis.org

1, 3, 18, 127, 999, 8376, 73400, 664143, 6157467, 58190531, 558478098, 5428532148, 53331912158, 528721992000, 5282688183600, 53140908294191, 537760961917833, 5470638540940401, 55914705172750446, 573908634206898951, 5913010265931479289, 61132102068652970100, 634002859944973526904

Offset: 1

Paul D. Hanna, Jun 25 2024

nonn

			G.f.: A(x) = x + 3*x^2 + 18*x^3 + 127*x^4 + 999*x^5 + 8376*x^6 + 73400*x^7 + 664143*x^8 + 6157467*x^9 + 58190531*x^10 + ...
where A( x*A(x)/(1 - A(x))^3 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 45*x^4 + 362*x^5 + 3084*x^6 + 27318*x^7 + 249149*x^8 + 2323968*x^9 + 22067697*x^10 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(2^n))^3 = x - 3*x^2 + 8*x^4 - 9*x^5 + 3*x^6 + 8*x^7 - 24*x^8 + 15*x^9 + 19*x^10 + ... + A373308(n-1) * x^n + ...
thus,
x = A(x) * (1 - A(x))^3 * (1 - A(x)^2)^3 * (1 - A(x)^4)^3 * (1 - A(x)^8)^3 * (1 - A(x)^16)^3 * ... * (1 - A(x)^(2^n))^3 * ...
Also, notice that the cube root of A(x)/x is an integral series
(A(x)/x)^(1/3) = 1 + x + 5*x^2 + 32*x^3 + 239*x^4 + 1937*x^5 + 16578*x^6 + 147408*x^7 + 1348465*x^8 + 12608851*x^9 + 119972595*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = (1/6) * Product_{n>=0} (1 - 1/6^(2^n))^3 = 0.0884290923082561345726735004152032422138677544...
A(t) = 1/7 at t = (1/7) * Product_{n>=0} (1 - 1/7^(2^n))^3 = 0.0844605844040460521136280418653467784637497846...
A(t) = 1/8 at t = (1/8) * Product_{n>=0} (1 - 1/8^(2^n))^3 = 0.0798174217593180496284155971364088109289815675...
A(1/12) = 0.1379538716718371951653031812720929490038524971492263...
A(1/13) = 0.1160657279647048938238673646663527089747582497393475...
A(1/14) = 0.1016889922856297159061963243507242491941351481713051...

Paul D. Hanna, Table of n, a(n) for n = 1..520

Cf. A372530, A373312, A106407, A373308, A373309.

{a(n) = my(A = serreverse(x*prod(k=0,#binary(n), (1 - x^(2^k) + x*O(x^n))^3))); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( subst(Ser(A),x, x*Ser(A)/(1 - Ser(A))^3 ) - Ser(A)^2,#A)); A[n+1]}
for(n=1,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x*A(x)/(1 - A(x))^3 ).
(2) A(x)^4 = A( x*A(x)^3/((1 - A(x))^3*(1 - A(x)^2)^3) ).
(3) A(x)^8 = A( x*A(x)^7/((1 - A(x))^3*(1 - A(x)^2)^3*(1 - A(x)^4)^3) ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1)/Product_{k=0..n-1} (1 - A(x)^(2^k))^3 ) for n > 0.
(5) A(x) = x / Product_{n>=0} (1 - A(x)^(2^n))^3.
(6) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n))^(3*(n+1)).
(7) A(x) = Series_Reversion( x*B(x) ), where B(x) = Product_{n>=0} (1 - x^(2^n))^3 is the g.f. of A373308.
The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 3*2^n * A(r)^(2^n)/(1 - A(r)^(2^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(2^n))^3, where r = 0.090173114826637655491436994778921911119292413640909... and A(r) = 0.197474208053634831172176658351098789075712647862486...
Given r and A(r) above, A(r) also satisfies 1 = Sum_{n>=0} 3*(n+1)*2^n * A(r)^(2^n)/(1 + A(r)^(2^n)).

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

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A373313 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^3 ).

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