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A199898 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum, and adjacent elements not both strictly positive and not both strictly negative.

%I A199898 #7 Jul 22 2025 13:26:27
%S A199898 1,1,3,1,5,7,1,7,15,15,1,9,25,49,33,1,11,37,111,159,75,1,13,51,209,
%T A199898 461,533,171,1,15,67,351,1043,2035,1783,391,1,17,85,545,2031,5725,
%U A199898 8823,6027,899,1,19,105,799,3573,13363,30199,39053,20437,2077,1,21,127,1121,5839
%N A199898 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum, and adjacent elements not both strictly positive and not both strictly negative.
%C A199898 Table starts
%C A199898 ....1.....1......1.......1........1........1.........1.........1..........1
%C A199898 ....3.....5......7.......9.......11.......13........15........17.........19
%C A199898 ....7....15.....25......37.......51.......67........85.......105........127
%C A199898 ...15....49....111.....209......351......545.......799......1121.......1519
%C A199898 ...33...159....461....1043.....2031.....3573......5839......9021......13333
%C A199898 ...75...533...2035....5725....13363....27457.....51395.....89577.....147547
%C A199898 ..171..1783...8823...30199....82555...193689....406575....783989....1413739
%C A199898 ..391..6027..39053..164993...536967..1462859...3500269...7584081...15191479
%C A199898 ..899.20437.172355..890299..3409609.10651367..28684325..68971571..151640029
%C A199898 .2077.69665.767425.4877477.22163661.80142549.245319361.661158741.1611184533
%H A199898 R. H. Hardin, <a href="/A199898/b199898.txt">Table of n, a(n) for n = 1..1096</a>
%F A199898 Empirical for rows:
%F A199898 T(1,k) = 1
%F A199898 T(2,k) = 2*k + 1
%F A199898 T(3,k) = k^2 + 5*k + 1
%F A199898 T(4,k) = (4/3)*k^3 + 6*k^2 + (20/3)*k + 1
%F A199898 T(5,k) = (11/12)*k^4 + (49/6)*k^3 + (193/12)*k^2 + (41/6)*k + 1
%F A199898 T(6,k) = (11/10)*k^5 + (55/6)*k^4 + (55/2)*k^3 + (173/6)*k^2 + (37/5)*k + 1
%F A199898 T(7,k) = (151/180)*k^6 + (163/15)*k^5 + (377/9)*k^4 + (395/6)*k^3 + (7429/180)*k^2 + (93/10)*k + 1
%e A199898 Some solutions for n=7 k=6
%e A199898 ..1....3....3....4...-3....4....4....0....3....3...-3...-6....4...-5....0....4
%e A199898 .-3...-4...-4...-6....5...-2...-6....6...-3...-5....0....3...-4....5...-3...-5
%e A199898 ..0....1....2....2...-3....4....4...-5....0....4....3...-1....0...-4....5....1
%e A199898 .-1....0...-3....0....0...-1...-3....3....4....0....0....6....5....6....0...-1
%e A199898 ..4....0....2....0....1....0....0...-6...-1....2...-2...-2...-6...-3...-3....5
%e A199898 .-6...-4...-2....3...-6...-5...-4....5....3....0....2....4....3....4....6....0
%e A199898 ..5....4....2...-3....6....0....5...-3...-6...-4....0...-4...-2...-3...-5...-4
%Y A199898 Column 1 is A136029(n+1)
%Y A199898 Row 3 is A082111
%K A199898 nonn,tabl
%O A199898 1,3
%A A199898 _R. H. Hardin_ Nov 11 2011