A188124 Number of strictly increasing arrangements of 5 nonzero numbers in -(n+3)..(n+3) with sum zero.

0, 4, 16, 42, 90, 172, 296, 482, 740, 1092, 1554, 2154, 2906, 3846, 4992, 6382, 8038, 10004, 12302, 14984, 18074, 21626, 25670, 30266, 35442, 41266, 47770, 55024, 63064, 71966, 81766, 92548, 104350, 117258, 131316, 146616, 163200, 181168, 200566

Offset: 0

Author

R. H. Hardin, Mar 21 2011

Keywords

• nonn

Comments

Row 5 of A188122.

Examples

			4*x + 16*x^2 + 42*x^3 + 90*x^4 + 172*x^5 + 296*x^6 + 482*x^7 + 740*x^8 + ...
Some solutions for n=6
.-7...-7...-6...-7...-8...-8...-4...-9...-7...-5...-6...-4...-6...-9...-7...-5
.-5...-5...-4...-6...-6...-2...-3...-5...-5...-4...-3...-3...-3...-5...-4...-3
..1....2....2....2....1...-1...-2....1...-4...-2...-2...-2....1....2...-2....1
..5....3....3....5....4....4....4....5....7....4....4....1....2....5....6....3
..6....7....5....6....9....7....5....8....9....7....7....8....6....7....7....4
		

Links

• R. H. Hardin, Table of n, a(n) for n = 0..200 (corrected by _R. H. Hardin_, Jan 19 2019)

Programs

PARI

{a(n) = local(v, c, m); m = n+3; forvec( v = vector( 5, i, [-m, m]), if( 0==prod( k=1, 5, v[k]), next); if( 0==sum( k=1, 5, v[k]), c++), 2); c} /* Michael Somos, Apr 11 2011 */

Formula

Empirical: a(n)=2*a(n-1)-a(n-3)-2*a(n-5)+2*a(n-6)+a(n-8)-2*a(n-10)+a(n-11) = 269/1728 +235*n^2/144 +161*n/96 +23*n^4/288 +83*n^3/144 +(-1)^n*(1/64-3*n/32) -2*(-1)^n*A130815(n+2)/27 +A057077(n+1)/8.

Empirical: G.f. -2*x*(2+4*x+5*x^2+5*x^3+4*x^4+x^5+2*x^6) / ( (x^2+1)*(1+x+x^2)*(1+x)^2*(x-1)^5 ). - R. J. Mathar, Mar 21 2011