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%I #16 Aug 27 2024 17:01:05
%S 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 25,55,73,58,24,5,1,9,32,86,141,151,94,33,5,1,10,41,126,252,338,289,
%U 151,43,6,1,11,50,177,414,676,734,526,227,55,6,1,12,61,241,649,1242,1656,1514
%N T(n,k) = number of nondecreasing arrangements of n numbers in -k..k with sum zero.
%H R. H. Hardin, <a href="/A183917/b183917.txt">Table of n, a(n) for n = 1..1350</a>
%H 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.
%e Table starts
%e 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 2 3 4 5 6 7 8 9 10 11 12 13 14
%e 2 5 8 13 18 25 32 41 50 61 72 85 98
%e 3 8 18 33 55 86 126 177 241 318 410 519 645
%e 3 12 32 73 141 252 414 649 967 1394 1944 2649 3523
%e 4 18 58 151 338 676 1242 2137 3486 5444 8196 11963 17002
%e 4 24 94 289 734 1656 3370 6375 11322 19138 30982 48417 73316
%e 5 33 151 526 1514 3788 8512 17575 33885 61731 107233 178870 288100
%e 5 43 227 910 2934 8150 20094 45207 94257 184717 343363 610358 1043534
%e 6 55 338 1514 5448 16660 44916 109583 246448 517971 1028172 1943488 3521260
%e Some solutions for n=5:
%e -2 -4 -4 -4 -4 -1 -4 -3 -4 -3 -1 -4 -3 -3 -2 -4
%e -2 0 0 -1 -2 0 -2 -2 -1 -3 -1 -4 0 -2 0 -3
%e 0 0 0 0 -1 0 1 -1 1 0 0 1 0 1 0 -1
%e 0 1 2 2 3 0 2 3 2 3 0 3 0 1 1 4
%e 4 3 2 3 4 1 3 3 2 3 2 4 3 3 1 4
%o (Python)
%o from sympy.utilities.iterables import partitions
%o 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 Column 2 is A001973.
%Y Column 3 is A001977.
%Y Column 4 is A001981.
%Y Diagonal is A109655.
%Y Row 3 is A000982(n+1).
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Jan 07 2011