A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 16, 18, 0, 0, 32, 100, 0, 0, 424, 510, 0, 0, 2792, 5988, 0, 0, 29058, 45106, 0, 0, 276828, 473854, 0, 0, 2455340, 4777436, 0, 0, 27466324, 46429640, 0, 0, 280395282, 526489336, 0, 0, 3193589950, 5661226928, 0, 0

Offset: 1

Author

Pietro Majer, Mar 19 2009

Keywords

• nonn

Comments

Constant term in the expansion of (x + 1/x)(x^16 + 1/x^16)..(x^n^4 + 1/x^n^4).

a(n)=0 for any n=1 (mod 4) or n=2 (mod 4).

Andrica & Tomescu give a more general integral formula than the one below. The asymptotic formula below is a conjecture by Andrica & Ionascu; it remains unproven. - Jonathan Sondow, Nov 11 2013

Examples

			For n=16 the a(16) = 2 solutions are +1 +16 +81 [phone] +4096 +6561 +10000 +14641 [phone] -50625 +65536 = 0 and the opposite.
		

Links

• D. Andrica and E. J. Ionascu, Variations on a result of Erdős and SurányiINTEGERS 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.

Crossrefs

Cf. A063865, A158092, A158118, A158380, A019568.

A111253(n) = a(n)/2. - Alois P. Heinz, Oct 31 2011

Programs

Maple

N:=32: p:=1 a:=[]: for n from 32 to N do p:=expand
(p*(x^(n^4)+x^(-n^4))): a:=[op(a), coeff(p,x,0)]: od:a;

Formula

Integral representation: a(n) = ((2^n)/Pi)*int_0^pi prod_{k=1}^n cos(x*k^4) dx.

Asymptotic formula: a(n) = (2^n)*sqrt(18/(Pi*n^9))*(1+o(1)) as n->infinity; n=-1 or 0 (mod 4).

Extensions

a(35)-a(58) from Alois P. Heinz, Oct 31 2011