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A096144
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Triangle T(n,k) = number of partitions of n in which the least part occurs exactly k times, k=1..n.
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4
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1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 4, 1, 1, 0, 1, 4, 3, 2, 1, 0, 1, 7, 3, 2, 1, 1, 0, 1, 8, 6, 2, 3, 1, 1, 0, 1, 12, 5, 6, 2, 2, 1, 1, 0, 1, 14, 11, 5, 4, 3, 2, 1, 1, 0, 1, 21, 11, 8, 5, 4, 2, 2, 1, 1, 0, 1, 24, 17, 11, 9, 4, 5, 2, 2, 1, 1, 0, 1, 34, 20, 15, 9, 8, 4, 4, 2, 2, 1, 1, 0, 1, 41, 30, 18, 14, 9, 7, 5, 4, 2, 2, 1, 1, 0, 1
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OFFSET
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1,4
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COMMENTS
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T(n,k) is the number of partitions of n for which the difference between the two largest distinct parts is k (in partitions having only 1 part, we assume that 0 is also a part). This follows easily from the definition by taking the conjugate partitions. Example: T(7,2) = 3 because we have [3,1,1,1,1], [3,3,1], and [4,2,1]. - Emeric Deutsch, Dec 05 2015.
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LINKS
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FORMULA
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G.f. for k-th column: sum(m>=1, x^(k*m)/prod(i>=m+1, 1-x^i ) ).
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EXAMPLE
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Triangle starts:
01: 1
02: 1 1
03: 2 0 1
04: 2 2 0 1
05: 4 1 1 0 1
06: 4 3 2 1 0 1
07: 7 3 2 1 1 0 1
08: 8 6 2 3 1 1 0 1
09: 12 5 6 2 2 1 1 0 1
10: 14 11 5 4 3 2 1 1 0 1
11: 21 11 8 5 4 2 2 1 1 0 1
12: 24 17 11 9 4 5 2 2 1 1 0 1
13: 34 20 15 9 8 4 4 2 2 1 1 0 1
14: 41 30 18 14 9 7 5 ...
T(7,2)=3 because we have: 5+1+1, 3+2+2, 3+2+1+1. - Geoffrey Critzer, Jun 20 2014
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MAPLE
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b:= proc(n, i) option remember; `if`(i=1, x^n,
`if`(irem(n, i, 'k')=0, x^k, 0)+
add(b(n-i*j, i-1), j=0..(n-1)/i))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
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MATHEMATICA
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nn=20; Table[Take[Map[Drop[#, 1]&, Drop[CoefficientList[Series[Sum[y x^k/(1-y x^k) Product[1/(1- x^j), {j, k+1, nn}], {k, 1, nn}], {x, 0, nn}], {x, y}], 1]][[i]], i], {i, 1, nn}]//Grid (* Geoffrey Critzer, Jun 20 2014 *)
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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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