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A183917 T(n,k) = number of nondecreasing arrangements of n numbers in -k..k with sum zero.

%I A183917 #20 May 12 2025 11:53:42
%S A183917 1,1,2,1,3,2,1,4,5,3,1,5,8,8,3,1,6,13,18,12,4,1,7,18,33,32,18,4,1,8,
%T A183917 25,55,73,58,24,5,1,9,32,86,141,151,94,33,5,1,10,41,126,252,338,289,
%U A183917 151,43,6,1,11,50,177,414,676,734,526,227,55,6,1,12,61,241,649,1242,1656,1514
%N A183917 T(n,k) = number of nondecreasing arrangements of n numbers in -k..k with sum zero.
%H A183917 R. H. Hardin, <a href="/A183917/b183917.txt">Table of n, a(n) for n = 1..1350</a>
%H A183917 David J. Hemmer and Karlee J. Westrem, <a href="https://arxiv.org/abs/2402.02250">Palindrome Partitions and the Calkin-Wilf Tree</a>, arXiv:2402.02250 [math.CO], 2024. See Definition 5.1, p. 8.
%H A183917 Karlee J. Westrem, <a href="https://digitalcommons.mtu.edu/etdr/1892/">Schaper numbers, palindrome partitions, and symmetric functions, with applications to characters of the symmetric group</a>, Ph. D. Dissertation, Michigan Tech. Univ. (2025). See p. 55.
%e A183917 Table starts
%e A183917  1  1   1    1    1     1     1      1      1      1       1       1       1
%e A183917  2  3   4    5    6     7     8      9     10     11      12      13      14
%e A183917  2  5   8   13   18    25    32     41     50     61      72      85      98
%e A183917  3  8  18   33   55    86   126    177    241    318     410     519     645
%e A183917  3 12  32   73  141   252   414    649    967   1394    1944    2649    3523
%e A183917  4 18  58  151  338   676  1242   2137   3486   5444    8196   11963   17002
%e A183917  4 24  94  289  734  1656  3370   6375  11322  19138   30982   48417   73316
%e A183917  5 33 151  526 1514  3788  8512  17575  33885  61731  107233  178870  288100
%e A183917  5 43 227  910 2934  8150 20094  45207  94257 184717  343363  610358 1043534
%e A183917  6 55 338 1514 5448 16660 44916 109583 246448 517971 1028172 1943488 3521260
%e A183917 Some solutions for n=5:
%e A183917   -2  -4  -4  -4  -4  -1  -4  -3  -4  -3  -1  -4  -3  -3  -2  -4
%e A183917   -2   0   0  -1  -2   0  -2  -2  -1  -3  -1  -4   0  -2   0  -3
%e A183917    0   0   0   0  -1   0   1  -1   1   0   0   1   0   1   0  -1
%e A183917    0   1   2   2   3   0   2   3   2   3   0   3   0   1   1   4
%e A183917    4   3   2   3   4   1   3   3   2   3   2   4   3   3   1   4
%o A183917 (Python)
%o A183917 from sympy.utilities.iterables import partitions
%o A183917 def A183917_T(n,k): return sum(1 for p in partitions(k*n,m=n,k=k<<1)) # _Chai Wah Wu_, Aug 27 2024
%Y A183917 Column 2 is A001973.
%Y A183917 Column 3 is A001977.
%Y A183917 Column 4 is A001981.
%Y A183917 Diagonal is A109655.
%Y A183917 Row 3 is A000982(n+1).
%K A183917 nonn,tabl
%O A183917 1,3
%A A183917 _R. H. Hardin_, Jan 07 2011