%I #18 Jan 19 2019 07:24:47

%S 0,4,16,42,90,172,296,482,740,1092,1554,2154,2906,3846,4992,6382,8038,

%T 10004,12302,14984,18074,21626,25670,30266,35442,41266,47770,55024,

%U 63064,71966,81766,92548,104350,117258,131316,146616,163200,181168,200566

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

%C Row 5 of A188122.

%H R. H. Hardin, <a href="/A188124/b188124.txt">Table of n, a(n) for n = 0..200</a> (corrected by _R. H. Hardin_, Jan 19 2019)

%F 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.

%F 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

%e 4*x + 16*x^2 + 42*x^3 + 90*x^4 + 172*x^5 + 296*x^6 + 482*x^7 + 740*x^8 + ...

%e Some solutions for n=6

%e .-7...-7...-6...-7...-8...-8...-4...-9...-7...-5...-6...-4...-6...-9...-7...-5

%e .-5...-5...-4...-6...-6...-2...-3...-5...-5...-4...-3...-3...-3...-5...-4...-3

%e ..1....2....2....2....1...-1...-2....1...-4...-2...-2...-2....1....2...-2....1

%e ..5....3....3....5....4....4....4....5....7....4....4....1....2....5....6....3

%e ..6....7....5....6....9....7....5....8....9....7....7....8....6....7....7....4

%o (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 */

%K nonn

%O 0,2

%A _R. H. Hardin_, Mar 21 2011