A111984 Expansion of g.f.: A(x) = ( Sum_{n>=0} (2*n+1)*8^n*x^(n*(n+1)/2) )^(1/3).

1, 8, -64, 960, -15360, 274432, -5223936, 103604224, -2115993600, 44188631040, -939004514304, 20235083907072, -441102457372672, 9708551114588160, -215436164616683520, 4814308934865944576, -108243166106365722624, 2446764246113433157632

Offset

0,2

Comments

Define F(x,q) = Sum_{n>=0} (2*n+1)*q^n*x^(n*(n+1)/2); then F(x,q)^(1/3) is an integer series in x when q = -1 (mod 9) or when q = 0 (mod 3). At q=-1 we have the famous result of Jacobi: F(x,-1)^(1/3) = (1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 +...)^(1/3) = 1 + Sum_{n>=1} (-1)^n * (x^(n*(3*n-1)/2) + x^(n*(3*n+1)/2) ).

Links

G. C. Greubel, Table of n, a(n) for n = 0..715

Formula

Conjecture: a(5*n+4) == 0 (mod 5) (checked up to n = 200). - Peter Bala, Feb 26 2021

Example

G.f.: A(x) = 1 + 8*x - 64*x^2 + 960*x^3 - 15360*x^4 + 274432*x^5 - 5223936*x^6 +...
where
A(x)^3 = 1 + 3*8*x + 5*8^2*x^3 + 7*8^3*x^6 + 9*8^4*x^10 + 11*8^5*x^15 + 13*8^6*x^21 + 15*8^7*x^28 + 17*8^8*x^36 + 19*8^9*x^45 + 21*8^10*x^55 +...

Maple

seq(coeff(series( ( add((2*n+1)*8^n*x^(n*(n+1)/2), n=0..40) )^(1/3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 05 2019

Mathematica

With[{nn=20}, CoefficientList[Series[Surd[Sum[(2n+1)8^n x^((n(n+1))/2), {n, 0, nn}], 3], {x, 0, nn}], x]] (* Harvey P. Dale, Jul 04 2013 *)

Prog

(PARI) a(n)=polcoeff(sum(k=0, sqrtint(2*n+1), (2*k+1)*8^k*x^(k*(k+1)/2)+x*O(x^n))^(1/3), n)
(Sage) [( (sum((2*n+1)*8^n*x^(n*(n+1)/2) for n in (0..40)) )^(1/3) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Nov 05 2019

Crossrefs

Cf. A111983 (g.f. A(x)^3), A111985 (g.f. A(x)^(1/4)).

Sequence in context: A087138 A293144 A349266 * A352721 A278145 A287230

Adjacent sequences: A111981 A111982 A111983 * A111985 A111986 A111987

Keyword

sign

Author

Paul D. Hanna, Aug 25 2005

Status

approved