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A188122 Table read by downward antidiagonals: T(n,k) is the number of strictly increasing arrangements of n nonzero numbers in -(n+k-2)..(n+k-2) with sum zero.

%I A188122 #15 Dec 14 2019 21:28:33
%S A188122 0,0,1,0,2,0,0,3,2,3,0,4,4,8,4,0,5,8,16,16,16,0,6,12,31,42,52,42,0,7,
%T A188122 18,51,90,137,152,137,0,8,24,80,172,308,426,484,426,0,9,32,118,296,
%U A188122 624,1032,1398,1536,1398,0,10,40,167,482,1154,2216,3528,4622,5064,4622,0,11,50
%N A188122 Table read by downward antidiagonals: T(n,k) is the number of strictly increasing arrangements of n nonzero numbers in -(n+k-2)..(n+k-2) with sum zero.
%C A188122 Table starts
%C A188122      0,    0,     0,     0,      0,      0,      0,      0,       0,       0,       0, ...
%C A188122      1,    2,     3,     4,      5,      6,      7,      8,       9,      10,      11, ...
%C A188122      0,    2,     4,     8,     12,     18,     24,     32,      40,      50,      60, ...
%C A188122      3,    8,    16,    31,     51,     80,    118,    167,     227,     302,     390, ...
%C A188122      4,   16,    42,    90,    172,    296,    482,    740,    1092,    1554,    2154, ...
%C A188122     16,   52,   137,   308,    624,   1154,   1999,   3278,    5144,    7772,   11387, ...
%C A188122     42,  152,   426,  1032,   2216,   4376,   8044,  13994,   23210,   37030,   57086, ...
%C A188122    137,  484,  1398,  3528,   7970,  16547,  32035,  58595,  102113,  170844,  275878, ...
%C A188122    426, 1536,  4622, 12124,  28660,  62222, 126122, 241250,  439514,  767656, 1292864, ...
%C A188122   1398, 5064, 15594, 42262, 103599, 233880, 493267, 982016, 1861168, 3379972, 5913676, ...
%H A188122 R. H. Hardin, <a href="/A188122/b188122.txt">Table of n, a(n) for n = 1..550</a>
%H A188122 Louis Ng, <a href="http://math.sfsu.edu/beck/teach/masters/louis.pdf">Magic counting with inside-out polytopes</a>, Master's Thesis, San Francisco State University, 2018.
%e A188122 Some solutions for n=8, k=6:
%e A188122   -11 -12 -11 -11 -12 -10 -11 -12 -12  -9 -10 -11 -11 -12 -12  -7
%e A188122    -9  -9 -10  -8  -6  -8  -9 -11  -9  -5  -8 -10  -8 -10 -10  -6
%e A188122    -5  -8  -4  -4  -4  -5  -8  -4  -8  -4  -5  -4  -7  -9  -5  -5
%e A188122    -4  -3  -2  -2  -1  -3  -3  -2   1  -2  -3  -1   1   1  -2  -3
%e A188122     2   5   2   1   2  -1   1  -1   4   1   4   3   2   2   3  -2
%e A188122     6   8   4   6   4   8   7   7   5   2   5   4   6   8   5   5
%e A188122     9   9  10   7   5   9  11  11   9   8   8   8   7   9   9   8
%e A188122    12  10  11  11  12  10  12  12  10   9   9  11  10  11  12  10
%Y A188122 Row 3 is A007590.
%K A188122 nonn,tabl
%O A188122 1,5
%A A188122 _R. H. Hardin_, Mar 21 2011