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A187080
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Triangle T(n,k) read by rows: fountains of n coins and height k.
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2
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1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 7, 1, 0, 0, 0, 0, 1, 12, 2, 0, 0, 0, 0, 0, 1, 20, 5, 0, 0, 0, 0, 0, 0, 1, 33, 11, 0, 0, 0, 0, 0, 0, 0, 1, 54, 22, 1, 0, 0, 0, 0, 0, 0, 0, 1, 88, 44, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 143, 85, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 232, 161, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 376, 302, 25, 0, 0, 0
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OFFSET
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0,13
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COMMENTS
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LINKS
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FORMULA
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T(n,1) + T(n,2) = Fibonacci(n).
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EXAMPLE
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Triangle begins:
1;
0,1;
0,1,0;
0,1,1,0;
0,1,2,0,0;
0,1,4,0,0,0;
0,1,7,1,0,0,0;
0,1,12,2,0,0,0,0;
0,1,20,5,0,0,0,0,0;
0,1,33,11,0,0,0,0,0,0;
0,1,54,22,1,0,0,0,0,0,0;
0,1,88,44,2,0,0,0,0,0,0,0;
0,1,143,85,5,0,0,0,0,0,0,0,0;
0,1,232,161,12,0,0,0,0,0,0,0,0,0;
0,1,376,302,25,0,0,0,0,0,0,0,0,0,0;
0,1,609,559,52,1,0,0,0,0,0,0,0,0,0,0;
0,1,986,1026,105,2,0,0,0,0,0,0,0,0,0,0,0;
0,1,1596,1870,207,5,0,0,0,0,0,0,0,0,0,0,0,0;
The 15 compositions corresponding to fountains of 7 coins are the following:
#: composition height
1: [ 1 2 3 1 ] 3
2: [ 1 2 2 2 ] 2
3: [ 1 1 2 3 ] 3
4: [ 1 2 2 1 1 ] 2
5: [ 1 2 1 2 1 ] 2
6: [ 1 1 2 2 1 ] 2
7: [ 1 2 1 1 2 ] 2
8: [ 1 1 2 1 2 ] 2
9: [ 1 1 1 2 2 ] 2
10: [ 1 2 1 1 1 1 ] 2
11: [ 1 1 2 1 1 1 ] 2
12: [ 1 1 1 2 1 1 ] 2
13: [ 1 1 1 1 2 1 ] 2
14: [ 1 1 1 1 1 2 ] 2
15: [ 1 1 1 1 1 1 1 ] 1
stats: 0 1 12 2 0 0 0 0
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MATHEMATICA
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b[n_, i_, h_] := b[n, i, h] = If[n == 0, x^h, Sum[b[n - j, j, Max[h, j]], {j, 1, Min[i + 1, n]}]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, 0, 0];
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CROSSREFS
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Row sums give A005169 (fountains of n coins).
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KEYWORD
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AUTHOR
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STATUS
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approved
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