%I #13 Feb 07 2019 02:34:02
%S 1,1,2,1,3,2,1,4,3,3,1,5,4,6,3,1,6,5,10,6,4,1,7,6,15,10,10,4,1,8,7,21,
%T 15,20,10,5,1,9,8,30,21,37,22,15,5,1,10,9,38,30,62,41,37,17,6,1,11,10,
%U 47,42,106,68,84,45,23,6,1,12,11,59,51,148,114,170,106,68,27,7,1,13,12,70,65
%N T(n,k) is the number of nondecreasing strings of numbers x(i=1..n) in -k..k with sum x(i)^3 equal to 0, read by antidiagonals.
%C Table starts:
%C .1..1..1...1...1....1....1....1....1.....1.....1.....1.....1.....1.....1......1
%C .2..3..4...5...6....7....8....9...10....11....12....13....14....15....16.....17
%C .2..3..4...5...6....7....8....9...10....11....12....13....14....15....16.....17
%C .3..6.10..15..21...30...38...47...59....70....82....99...113...128...144....163
%C .3..6.10..15..21...30...42...51...65....78....92...111...129...152...172....193
%C .4.10.20..37..62..106..148..197..280...366...470...637...778...922..1098...1327
%C .4.10.22..41..68..114..202..273..402...548...720...979..1248..1660..2072...2525
%C .5.15.37..84.170..346..552..817.1319..1951..2817..4262..5776..7388..9688..12753
%C .5.17.45.106.216..422..890.1415.2401..3809..5725..8810.12622.18662.26200..35595
%C .6.23.68.186.450.1070.2020.3505.6456.10987.18010.30214.46352.66586.98330.142605
%H R. H. Hardin, <a href="/A188277/b188277.txt">Table of n, a(n) for n = 1..1196</a>
%F T(n,k) = [y^0] [x^n] Product_{j=-k..k} 1/(1-x*y^(j^3)). - _Robert Israel_, Feb 06 2019
%e Some solutions for n=9, k=8:
%e -8 -6 -8 -6 -8 -6 -8 -7 -8 -7 -8 -5 -5 -7 -7 -7
%e -4 -6 -7 -5 -6 -6 -6 -5 -7 -7 -1 -2 -1 -2 -7 -4
%e -1 -4 -2 -3 -6 -6 -3 -3 -6 -4 -1 -2 -1 -1 -4 -3
%e 0 -3 -1 -1 -4 -3 0 -2 -5 -4 2 0 -1 0 1 1
%e 0 -3 2 -1 -2 0 0 -2 -1 0 2 0 -1 0 1 1
%e 0 -2 4 -1 -2 3 0 -1 -1 4 3 0 0 2 4 3
%e 1 -1 4 1 0 6 3 -1 7 4 4 2 1 4 5 4
%e 4 6 6 3 8 6 6 1 7 7 4 2 4 4 6 5
%e 8 7 8 7 8 6 8 8 8 7 7 5 4 6 7 6
%p T:= proc(n,k) local P,S,j;
%p P:= 1/mul(1-x*y^(j^3),j=-k..k);
%p S:= series(P,x,n+1);
%p coeff(coeff(S,x,n),y,0)
%p end proc:
%p seq(seq(T(i,m-i),i=1..m-1),m=2..16); # _Robert Israel_, Feb 06 2019
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Mar 26 2011