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A027293
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Triangular array given by rows: P(n,k) = number of partitions of n that contain k as a part.
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19
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1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 3, 2, 1, 1, 7, 5, 3, 2, 1, 1, 11, 7, 5, 3, 2, 1, 1, 15, 11, 7, 5, 3, 2, 1, 1, 22, 15, 11, 7, 5, 3, 2, 1, 1, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 77
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| A027293 * an infinite lower triangular matrix with A010815 (1, -1, -1, 0, 0, 1,...) as the main diagonal the rest zeros = triangle A145975 having row sums = [1, 0, 0, 0,...]. These matrix operations are equivalent to the comment in A010815 stating "when convolved with the partition numbers = [1, 0, 0, 0,...]. [From Gary W. Adamson, Oct 25 2008]
Contribution from Gary W. Adamson, Oct 26 2008: (Start)
Row sums = A000070: (1, 2, 4, 7, 12, 19, 30, 45, 67,...)
A027293^2 = triangle A146023 (End)
P(n,k) is also the number of partitions of n-1 that contain k-1 as a part: P(n,k) = P(n-1,k-1) for k>1. [From Omar E. Pol, Nov 29 2010]
Triangle read by rows in which row n lists the first n partition numbers A000041 in decreasing order. - Omar E. Pol, Aug 06 2011
1) It appears that P(n,k) is also the total number of occurrences of k in the last k shells of the partitions of n (Cf. A135010, A182703). 2) It appears that P(n,k) is also the difference, between n and n-k, of the total number of occurrences of k in all their partitions (Cf. A066633). - Omar E. Pol, Feb 07 2012
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EXAMPLE
| Triangle begins:
1
1 1
2 1 1
3 2 1 1
5 3 2 1 1
7 5 3 2 1 1
11 7 5 3 2 1 1
15 11 7 5 3 2 1 1
22 15 11 7 5 3 2 1 1
30 22 15 11 7 5 3 2 1 1
42 30 22 15 11 7 5 3 2 1 1
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MATHEMATICA
| Needs["DiscreteMath`Combinatorica`"]
f[n_] := Block[{t = Flatten[Union /@ Partitions@n]}, Table[Count[t, i], {i, n}]]; Array[f, 13] // Flatten.
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CROSSREFS
| Every column of P is A000041.
A145975, A010815 [From Gary W. Adamson, Oct 25 2008]
A000070, A146023 [From Gary W. Adamson, Oct 26 2008]
A182700 [From Omar E. Pol, Nov 29 2010]
Sequence in context: A114282 A112739 A194543 * A104762 A152462 A180360
Adjacent sequences: A027290 A027291 A027292 * A027294 A027295 A027296
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KEYWORD
| nonn,tabl,changed
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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