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A188277
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T(n,k) is the number of nondecreasing strings of numbers x(i=1..n) in -k..k with sum x(i)^3 equal to 0, read by antidiagonals.
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13
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1, 1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 6, 3, 1, 6, 5, 10, 6, 4, 1, 7, 6, 15, 10, 10, 4, 1, 8, 7, 21, 15, 20, 10, 5, 1, 9, 8, 30, 21, 37, 22, 15, 5, 1, 10, 9, 38, 30, 62, 41, 37, 17, 6, 1, 11, 10, 47, 42, 106, 68, 84, 45, 23, 6, 1, 12, 11, 59, 51, 148, 114, 170, 106, 68, 27, 7, 1, 13, 12, 70, 65
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OFFSET
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1,3
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COMMENTS
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Table starts:
.1..1..1...1...1....1....1....1....1.....1.....1.....1.....1.....1.....1......1
.2..3..4...5...6....7....8....9...10....11....12....13....14....15....16.....17
.2..3..4...5...6....7....8....9...10....11....12....13....14....15....16.....17
.3..6.10..15..21...30...38...47...59....70....82....99...113...128...144....163
.3..6.10..15..21...30...42...51...65....78....92...111...129...152...172....193
.4.10.20..37..62..106..148..197..280...366...470...637...778...922..1098...1327
.4.10.22..41..68..114..202..273..402...548...720...979..1248..1660..2072...2525
.5.15.37..84.170..346..552..817.1319..1951..2817..4262..5776..7388..9688..12753
.5.17.45.106.216..422..890.1415.2401..3809..5725..8810.12622.18662.26200..35595
.6.23.68.186.450.1070.2020.3505.6456.10987.18010.30214.46352.66586.98330.142605
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LINKS
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FORMULA
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T(n,k) = [y^0] [x^n] Product_{j=-k..k} 1/(1-x*y^(j^3)). - Robert Israel, Feb 06 2019
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EXAMPLE
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Some solutions for n=9, k=8:
-8 -6 -8 -6 -8 -6 -8 -7 -8 -7 -8 -5 -5 -7 -7 -7
-4 -6 -7 -5 -6 -6 -6 -5 -7 -7 -1 -2 -1 -2 -7 -4
-1 -4 -2 -3 -6 -6 -3 -3 -6 -4 -1 -2 -1 -1 -4 -3
0 -3 -1 -1 -4 -3 0 -2 -5 -4 2 0 -1 0 1 1
0 -3 2 -1 -2 0 0 -2 -1 0 2 0 -1 0 1 1
0 -2 4 -1 -2 3 0 -1 -1 4 3 0 0 2 4 3
1 -1 4 1 0 6 3 -1 7 4 4 2 1 4 5 4
4 6 6 3 8 6 6 1 7 7 4 2 4 4 6 5
8 7 8 7 8 6 8 8 8 7 7 5 4 6 7 6
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MAPLE
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T:= proc(n, k) local P, S, j;
P:= 1/mul(1-x*y^(j^3), j=-k..k);
S:= series(P, x, n+1);
coeff(coeff(S, x, n), y, 0)
end proc:
seq(seq(T(i, m-i), i=1..m-1), m=2..16); # Robert Israel, Feb 06 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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