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A089177
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Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n)) giving number of non-squashing partitions of n into k parts.
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3
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1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 4, 1, 1, 5, 6, 2, 1, 6, 9, 4, 1, 7, 12, 6, 1, 8, 16, 10, 1, 1, 9, 20, 14, 2, 1, 10, 25, 20, 4, 1, 11, 30, 26, 6, 1, 12, 36, 35, 10, 1, 13, 42, 44, 14, 1, 14, 49, 56, 20, 1, 15, 56, 68, 26, 1, 16, 64, 84, 36, 1, 1, 17, 72, 100, 46, 2, 1, 18, 81, 120, 60, 4, 1
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OFFSET
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0,5
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LINKS
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Alois P. Heinz, Rows n = 0..1002, flattened
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.
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FORMULA
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Row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor(n/2) shifted one place right).
G.f. for column k (k >= 2): x^(2^(k-2))/((1-x)*Product_j=1..k-2} (1-x^(2^j))).
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 2;
1, 4, 4, 1;
1, 5, 6, 2;
1, 6, 9, 4;
1, 7, 12, 6;
1, 8, 16, 10, 1;
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MAPLE
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T:= proc(n) option remember;
`if`(n=0, 1, zip((x, y)-> x+y, [T(n-1)], [0, T(floor(n/2))], 0)[])
end:
seq (T(n), n=0..25); # Alois P. Heinz, Apr 01 2012
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CROSSREFS
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Cf. A089178. Columns give A002620, A008804, A088932, A088954. Row sums give A000123.
Sequence in context: A094363 A124832 A137569 * A023996 A049998 A029253
Adjacent sequences: A089174 A089175 A089176 * A089178 A089179 A089180
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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N. J. A. Sloane, Dec 08 2003
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EXTENSIONS
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More terms from Alford Arnold, May 22 2004
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STATUS
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approved
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