A158380 Number of solutions to +-1 +- 3 +- 6 +- ... +- n(n+1)/2 = 0.

1, 0, 0, 0, 2, 0, 2, 2, 4, 0, 12, 16, 26, 0, 66, 104, 210, 0, 620, 970, 1748, 0, 5948, 10480, 18976, 0, 60836, 111430, 209460, 0, 704934, 1284836, 2387758, 0, 8331820, 15525814, 28987902, 0, 101242982, 190267598, 358969426, 0, 1275032260, 2404124188, 4547419694

Offset: 0

Author

Pietro Majer, Mar 17 2009

Keywords

• easy
• nonn

Comments

Equivalently, number of partitions of the set of the first n triangular numbers {t(1),...,t(n)} into two classes with equal sums.

Constant term in the expansion of (x + 1/x)(x^3 + 1/x^3)...(x^t(n) + 1/x^t(n)).

a(n) = 0 for all n == 1 (mod 4).

Andrica & Tomescu give a more general integral formula than the one below. - Jonathan Sondow, Nov 11 2013

Examples

			For n=6 the 2 solutions are +1-3+6-10-15+21 = 0 and -1+3-6+10+15-21 = 0.
		

Links

• Ray Chandler, Table of n, a(n) for n = 0..633
• 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. A058498, A063865, A158092, A158118.

Programs

Maple

N:=70: p:=1: a:=[]: for n from 0 to N do
p:=expand(p*(x^(n*(n+1)/2)+x^(-n*(n+1)/2))):
a:=[op(a), coeff(p, x, 0)]: od:a;
# second Maple program:
b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
      `if`(n=m, 1, b(abs(n-i*(i+1)/2), i-1)+
      b(n+i*(i+1)/2, i-1))))((2+(3+i)*i)*i/6)
    end:
a:= n-> `if`(irem(n, 4)=1, 0, b(0, n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 17 2017

Mathematica

a[n_] := With[{t = Table[k(k+1)/2, {k, 1, n}]}, Coefficient[Times @@ (x^t + 1/x^t), x, 0]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 50}] (* Jean-François Alcover, Jun 16 2018 *)

PARI

t(k) = k*(k+1)/2;
a(n) = polcoeff(prod(k=1, n, (x^t(k)+ 1/x^t(k))), 0); \\ Michel Marcus, May 19 2015

Formula

a(n) = (2^n/Pi) * Integral_{x=0..Pi} cos(x)*cos(3x)*...*cos(n(n+1)x/2) dx.

a(n) ~ 2^(n+1)*sqrt(10/Pi)*n^(-5/2)*(1+o(1)) as n --> infinity, n !== 1 (mod 4).

a(n) = 2 * A058498(n) for n > 0. - Alois P. Heinz, Nov 01 2011

Extensions

a(0) = 1 prepended by Joerg Arndt, Sep 17 2017

Example corrected by Ilya Gutkovskiy, Feb 02 2022