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A158118 Number of solutions of +-1+-2^3+-3^3..+-n^3=0.

%I A158118 #26 Dec 13 2014 00:48:39
%S A158118 0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,2,0,0,4,124,0,0,536,712,0,0,4574,2260,
%T A158118 0,0,10634,73758,0,0,406032,638830,0,0,4249160,3263500,0,0,21907736,
%U A158118 82561050,0,0,485798436,945916970,0,0,5968541478,6839493576,0,0
%N A158118 Number of solutions of +-1+-2^3+-3^3..+-n^3=0.
%C A158118 Constant term in the expansion of (x + 1/x)(x^8 + 1/x^8)..(x^n^3 + 1/x^n^3).
%C A158118 a(n) = 0 for any n=1 (mod 4) or n=2 (mod 4).
%C A158118 The expansion above and the integral representation formula below are due to Andrica & Tomescu. The asymptotic formula is a conjecture; see Andrica & Ionascu. - _Jonathan Sondow_, Nov 06 2013
%H A158118 Ray Chandler, <a href="/A158118/b158118.txt">Table of n, a(n) for n = 1..130</a>
%H A158118 D. Andrica and E. J. Ionascu, <a href="http://www.dorinandrica.ro/files/presentation-INTEGERS-2013.pdf">Variations on a result of Erdős and Surányi</a>, INTEGERS 2013 slides.
%H A158118 Dorin Andrica and Ioan Tomescu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Tomescu/tomescu4.html">On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance</a>, J. Integer Sequences, 5 (2002), Article 02.2.4.
%F A158118 a(n) = 2 * A113263(n).
%F A158118 Integral representation: a(n)=((2^n)/Pi)*int_0^Pi prod_{k=1}^n cos(x*k^3) dx.
%F A158118 Asymptotic formula: a(n)=(2^n)*sqrt(14/(Pi*n^7))*(1+o(1)) as n-->infty; n=-1 or 0 (mod 4).
%e A158118 Example: For n=12 the a(12) = 2 solutions are:
%e A158118 +1+8-27+64-125-216-343+512+729-1000-1331+1728=0,
%e A158118 -1-8+27-64+125+216+343-512-729+1000+1331-1728=0.
%p A158118 N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*( x^(n^3) + x^(-n^3) )): a:=[op(a), coeff(p,x,0)]: od:a;
%Y A158118 Equals twice A113263.
%Y A158118 Cf. A063865, A158092, A019568. - _Pietro Majer_, Mar 15 2009
%K A158118 nonn
%O A158118 1,12
%A A158118 _Pietro Majer_, Mar 12 2009