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A060043
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Triangle T(n,k), n >= 1, k >= 1, of generalized sum of divisors function, read by rows.
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6
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1, 3, 1, 4, 3, 7, 9, 6, 1, 15, 12, 3, 30, 8, 9, 45, 15, 22, 67, 13, 1, 42, 99, 18, 3, 81, 135, 12, 9, 140, 175, 28, 22, 231, 231, 14, 51, 351, 306, 24, 1, 97, 551, 354, 24, 3, 188, 783, 465, 31, 9, 330, 1134, 540, 18, 22, 568, 1546, 681, 39, 51, 918, 2142, 765, 20
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Lengths of rows are 1 1 2 2 2 3 3 3 3 ... (A003056).
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REFERENCES
| P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
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FORMULA
| T(n, 1) = sum of divisors of n (A000203), T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*m_1 + s_2*m_2 + ... + s_k*m_k = n and the sum is over all such k-partitions of n.
G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1-q^m_1)*(1-q^m_2)*...*(1-q^m_k))^2 = Sum_n T(n, k)*q^n.
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EXAMPLE
| Triangle turned on its side begins:
1 3 4 7 6 12 .8 15 13 18 etc
. . 1 3 9 15 30 45 67 99 etc
. . . . . .1 .3 .9 22 42 etc
. . . . . . . . .. .. .1 etc
For example, T(6,2) = 15.
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CROSSREFS
| Diagonals give A000203, A002127, A002128. Cf. A060044.
Sequence in context: A054019 A035626 A082587 * A054907 A089554 A129246
Adjacent sequences: A060040 A060041 A060042 * A060044 A060045 A060046
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KEYWORD
| nonn,tabf,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mar 19 2001
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EXTENSIONS
| More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 24 2002
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