A370145 Expansion of ( (1 + x)*(1 - 2*x)*(1 + 4*x) )^(1/3).

1, 1, -3, 3, -12, 30, -102, 318, -1083, 3657, -12747, 44715, -159222, 571332, -2068608, 7538664, -27646374, 101915850, -377496030, 1404077790, -5242135728, 19637862132, -73793090676, 278068062756, -1050503580534, 3977985415746, -15096209345958, 57403753019238, -218683959367908

Offset

0,3

Comments

The cube root of F(x) = (1 + x)*(1 - 2*x)*(1 + 4*x) = (1 + 3*x - 6*x^2 - 8*x^3) is an integer series because F(x) == (1+x)^3 (mod 9).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..630

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x)^3 = (1 + x)*(1 - 2*x)*(1 + 4*x) = (1 + 3*x - 6*x^2 - 8*x^3).

(2) Product_{n>=1} A( 2^(n-1)*x^n )^3 = Sum_{n>=0} 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2), which is the g.f. of A370015.

(3) A(x) = 1/B(x/A(x)) where B(x) = 1/A(x/B(x)) = x/Series_Reversion(x/A(x)) equals the g.f. of A370146.

a(n) ~ (-1)^(n+1) * 2^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 23 2024

Example

G.f.: A(x) = 1 + x - 3*x^2 + 3*x^3 - 12*x^4 + 30*x^5 - 102*x^6 + 318*x^7 - 1083*x^8 + 3657*x^9 - 12747*x^10 + 44715*x^11 - 159222*x^12 + ...
where A(x)^3 = (1 + 3*x - 6*x^2 - 8*x^3).
RELATED SERIES.
The following infinite product equals the g.f. of A370015:
A(x)^3 * A(2*x^2)^3 * A(4*x^3)^3 * A(8*x^4)^3 * ... * A(2^(n-1)*x^n)^3 * ... = 1 + 3*x + 22*x^3 + 344*x^6 + 10944*x^10 + 699392*x^15 + 89489408*x^21 + 22907191296*x^28 + ... + 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) + ... by the Jacobi triple product identity.
If A(x) = 1/B(x/A(x)) then B(x) = 1/A(x/B(x)) begins
B(x) = 1 - x + 3*x^2 - 9*x^6 + 27*x^8 - 324*x^12 + 1215*x^14 - 18711*x^18 + 75816*x^20 - 1301265*x^24 + 5484996*x^26 - 100048689*x^30 + 431943435*x^32 - 8192222064*x^36 + ... + A370146(n)*x^n + ...

Prog

(PARI) {a(n) = polcoeff( (1 + 3*x - 6*x^2 - 8*x^3 +x*O(x^n))^(1/3), n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A370015, A370016, A370146, A370147, A370149.

Sequence in context: A328150 A161804 A097342 * A025236 A014432 A107330

Adjacent sequences: A370142 A370143 A370144 * A370146 A370147 A370148

Keyword

sign

Author

Paul D. Hanna, Feb 23 2024

Status

approved