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A145518
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Triangle read by rows: T1[n,k;x] := sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040
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2
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2, 3, 4, 5, 6, 8, 7, 19, 12, 16, 11, 29, 38, 24, 32, 13, 68, 85, 76, 48, 64, 17, 94, 181, 170, 152, 96, 128, 19, 177, 326, 443, 340, 304, 192, 256, 23, 231, 683, 787, 886, 680, 608, 384, 512, 29, 400, 1066, 1780, 1817, 1772, 1360, 1216, 768, 1024, 31, 484, 1899, 3119
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Let p(n; m_1, m_2, m_3, ..., m_n) denote a partition of integer n in exponential representation, i.e. the m_i are the counts of parts i and satisfy 1*m_1 + 2*m_2 + 3*m_3 + ... + n*m_n = n.
Let p(n, k; m_1, m_2, m_3, ..., m_n) be the partitions of n into exactly k parts; these are further constrained by m_1 + m_2 + m_3 + ... + m_n = k.
Then the triangle is given by T1[n,k;x] := sum_{all p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n,
where x_i is the i-th prime number (A000040).
2nd column (4,6,19,29,68,94,177, ...) is A024697.
Row sums give A145519.
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LINKS
| Tilman Neumann, More terms, partition generator and transform implementation.
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CROSSREFS
| cf. A000040, A024697, A145519, A145520
Sequence in context: A129129 A114622 A125624 * A130916 A003965 A097502
Adjacent sequences: A145515 A145516 A145517 * A145519 A145520 A145521
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KEYWORD
| nonn,tabl
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AUTHOR
| Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 12 2008
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EXTENSIONS
| Changed reference to more terms etc. to make it version independent Tilman Neumann (Tilman.Neumann(AT)web.de), Sep 02 2009
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