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%I #17 Feb 27 2021 21:29:45
%S 1,8,-64,960,-15360,274432,-5223936,103604224,-2115993600,44188631040,
%T -939004514304,20235083907072,-441102457372672,9708551114588160,
%U -215436164616683520,4814308934865944576,-108243166106365722624,2446764246113433157632
%N Expansion of g.f.: A(x) = ( Sum_{n>=0} (2*n+1)*8^n*x^(n*(n+1)/2) )^(1/3).
%C 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) ).
%H G. C. Greubel, <a href="/A111984/b111984.txt">Table of n, a(n) for n = 0..715</a>
%F Conjecture: a(5*n+4) == 0 (mod 5) (checked up to n = 200). - _Peter Bala_, Feb 26 2021
%e G.f.: A(x) = 1 + 8*x - 64*x^2 + 960*x^3 - 15360*x^4 + 274432*x^5 - 5223936*x^6 +...
%e where
%e 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 +...
%p 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
%t 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 *)
%o (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)
%o (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
%Y Cf. A111983 (g.f. A(x)^3), A111985 (g.f. A(x)^(1/4)).
%K sign
%O 0,2
%A _Paul D. Hanna_, Aug 25 2005
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