A188125 Number of strictly increasing arrangements of 6 nonzero numbers in -(n+4)..(n+4) with sum zero.

4, 16, 52, 137, 308, 624, 1154, 1999, 3278, 5144, 7772, 11387, 16230, 22602, 30830, 41303, 54440, 70734, 90706, 114963, 144146, 178984, 220244, 268797, 325548, 391514, 467756, 555449, 655816, 770208, 900020, 1046787, 1212094, 1397668

Offset: 0

Author

R. H. Hardin, Mar 21 2011

Keywords

• nonn

Comments

Row 6 of A188122.

Examples

			4 + 16*x + 52*x^2 + 137*x^3 + 308*x^4 + 624*x^5 + 1154*x^6 + 1999*x^7 + 3278*x^8 + ...
Some solutions for n=6
-10...-8...-7...-8...-8...-9...-9...-9...-9...-7..-10...-9...-7..-10...-9...-9
.-8...-6...-5...-5...-6...-3...-7...-3...-2...-5...-6...-5...-5...-6...-4...-5
.-1....1...-1...-1...-1...-2...-2....1...-1...-2...-2...-1...-1...-2...-2...-4
..4....3....1....1....2....3....3....2....1....1....2....1....3....3....1....3
..7....4....2....3....5....4....5....4....2....4....6....6....4....5....5....6
..8....6...10...10....8....7...10....5....9....9...10....8....6...10....9....9
		

Links

• R. H. Hardin, Table of n, a(n) for n = 0..200 (a(0) inserted by _Georg Fischer_, Jan 24 2019)

Crossrefs

Cf. A188122

Programs

PARI

{a(n) = local(v, c, m); m = n+4; forvec( v = vector( 6, i, [-m, m]), if( 0==prod( k=1, 6, v[k]), next); if( 0==sum( k=1, 6, v[k]), c++), 2); c} /* Michael Somos, Apr 11 2011 */

Formula

Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-5)+2*a(n-8)-a(n-11)-a(n-13)+2*a(n-15)-a(n-16)

= 168587/43200 +187*n/32 +3593*n^3/2160 +619*n^2/144 +457*n^4/1440 +11*n^5/450 -(-1)^n/64-3*n*(-1)^n/32 +4*(-1)^n*A119910(n+1)/27 -2*A117444(n+2)/25 +A057077(n)/8.

Empirical: G.f. -x*(-16 -20*x -33*x^2 -50*x^3 -60*x^4 -59*x^5 -51*x^6 -41*x^7 -18*x^8 -3*x^9 -x^10 +x^11 +4*x^12 -2*x^13 -7*x^14 +4*x^15) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(x^2+1) *(1+x)^2 *(x-1)^6 ). - R. J. Mathar, Mar 21 2011