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A126442
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Triangular array t read by rows: t(0,k) is p(k), the number of partitions of the k-multiset {0,0,...,0} with k zeros. For 0 <= n < k, t(n, k) is the number of partitions of the k-multiset {0, 0, ..., 0, 1, 2, 3, ..., k-n} with n zeros.
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7
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1, 2, 2, 3, 4, 5, 5, 7, 11, 15, 7, 12, 21, 36, 52, 11, 19, 38, 74, 135, 203, 15, 30, 64, 141, 296, 566, 877, 22, 45, 105, 250, 592, 1315, 2610, 4140, 30, 67, 165, 426, 1098, 2752, 6393, 13082, 21147, 42, 97, 254, 696, 1940, 5317, 13960, 33645, 70631, 115975
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| First in a series of triangular arrays which comprise subsequences of A096443(n).
The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.
Let t(n, k) be the number of ways to partition the k-multiset {0,0,...,0,1,2,3,4,...,k-n} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i, j) p(k - n - i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.
To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(k-n-i) ways to partition the remaining k-n-i zeros. Multiplying and summing gives the result. - George Beck, Jan 10 2011
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EXAMPLE
| This first array includes only the hook cases. A096443(9,14,16) correspond to partitions [2,2], [3,2] and [2,2,1] so these values do not appear in A126442.
The array begins:
1
2 2
3 4 5
5 7 11 15
7 12 21 36 52
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MATHEMATICA
| (* The triangle is flattened to a sequence. *) Get["Combinatorica`"]; t[n_, k_] := Sum[StirlingS2[n, j] * NumberOfCompositions[i, j] * PartitionsP[k - n - i], {j, 0, n}, {i, 0, k - n}]; Table[ t[n, k], {k, 10}, {n, 0, k - 1}] // Flatten (* George Beck, Jan 10 2011 *)
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CROSSREFS
| Cf. A000041, A000070, A082775, A093802, A000291, A002763, A000412, A054225, A035310, A000110, A035098.
Sequence in context: A015743 A015755 A096443 * A129306 A114094 A179255
Adjacent sequences: A126439 A126440 A126441 * A126443 A126444 A126445
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KEYWORD
| nonn,tabl
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AUTHOR
| Alford Arnold (Alford1940(AT)aol.com), Jan 28 2007
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EXTENSIONS
| Definition clarified by George Beck (george.beck(AT)gmail.com), Jan 11 2011
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