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A188181 T(n,k) is the number of strictly increasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.

%I A188181 #10 Apr 18 2025 09:53:37
%S A188181 1,1,1,1,2,2,1,3,5,5,1,4,8,12,12,1,5,13,24,32,32,1,6,18,43,73,94,94,1,
%T A188181 7,25,69,141,227,289,289,1,8,32,104,252,480,734,910,910,1,9,41,150,
%U A188181 414,920,1656,2430,2934,2934,1,10,50,207,649,1636,3370,5744,8150,9686,9686,1,11
%N A188181 T(n,k) is the number of strictly increasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.
%H A188181 R. H. Hardin, <a href="/A188181/b188181.txt">Table of n, a(n) for n = 1..550</a>
%F A188181 T(3,n) = A000982(n+1).
%e A188181 Table starts
%e A188181 ....1....1.....1.....1......1......1......1.......1.......1.......1.......1
%e A188181 ....1....2.....3.....4......5......6......7.......8.......9......10......11
%e A188181 ....2....5.....8....13.....18.....25.....32......41......50......61......72
%e A188181 ....5...12....24....43.....69....104....150.....207.....277.....362.....462
%e A188181 ...12...32....73...141....252....414....649.....967....1394....1944....2649
%e A188181 ...32...94...227...480....920...1636...2739....4370....6698....9926...14293
%e A188181 ...94..289...734..1656...3370...6375..11322...19138...30982...48417...73316
%e A188181 ..289..910..2430..5744..12346..24591..46029...81805..139143..227930..361384
%e A188181 ..910.2934..8150.20094..45207..94257.184717..343363..610358.1043534.1724882
%e A188181 .2934.9686.27718.70922.165821.360002.734517.1421530.2628824.4672836.8022362
%e A188181 Some solutions for n=7 and k=5:
%e A188181 .-7...-9...-8..-10...-6...-6...-9...-8...-8...-7...-9..-10...-9...-8...-9...-7
%e A188181 .-5...-7...-6...-7...-5...-5...-4...-6...-7...-3...-8...-5...-3...-7...-5...-4
%e A188181 .-3...-1...-5...-4...-4...-3...-1...-4...-5...-2...-6...-3...-1...-3...-4...-3
%e A188181 .-1....0...-1....3...-3...-2....0...-1....0...-1....4....0....1....1...-1...-1
%e A188181 ..3....3....2....5...-1....1....1....4....3....0....5....3....3....2....4....0
%e A188181 ..4....4....8....6....9....7....4....5....7....4....6....5....4....5....6....7
%e A188181 ..9...10...10....7...10....8....9...10...10....9....8...10....5...10....9....8
%Y A188181 Column 1 is A076822.
%Y A188181 Column 2 is A002838.
%Y A188181 Cf. A000982.
%K A188181 nonn,tabl
%O A188181 1,5
%A A188181 _R. H. Hardin_, Mar 23 2011