

A158118


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


6



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, 0, 0, 10634, 73758, 0, 0, 406032, 638830, 0, 0, 4249160, 3263500, 0, 0, 21907736, 82561050, 0, 0, 485798436, 945916970, 0, 0, 5968541478, 6839493576, 0, 0
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OFFSET

1,12


COMMENTS

Constant term in the expansion of (x + 1/x)(x^8 + 1/x^8)..(x^n^3 + 1/x^n^3).
a(n) = 0 for any n=1 (mod 4) or n=2 (mod 4).
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


LINKS

Ray Chandler, Table of n, a(n) for n = 1..130
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.


FORMULA

a(n) = 2 * A113263(n).
Integral representation: a(n)=((2^n)/Pi)*int_0^Pi prod_{k=1}^n cos(x*k^3) dx.
Asymptotic formula: a(n)=(2^n)*sqrt(14/(Pi*n^7))*(1+o(1)) as n>infty; n=1 or 0 (mod 4).


EXAMPLE

Example: For n=12 the a(12) = 2 solutions are:
+1+827+64125216343+512+72910001331+1728=0,
18+2764+125+216+343512729+1000+13311728=0.


MAPLE

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;


CROSSREFS

Equals twice A113263.
Cf. A063865, A158092, A019568.  Pietro Majer, Mar 15 2009
Sequence in context: A118965 A252729 A121552 * A212137 A346462 A230295
Adjacent sequences: A158115 A158116 A158117 * A158119 A158120 A158121


KEYWORD

nonn


AUTHOR

Pietro Majer, Mar 12 2009


STATUS

approved



