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Search: id:A060177
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| A060177 |
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Triangle of generalized sum of divisors function, read by rows. |
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+0
10
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| 1, 2, 1, 2, 2, 3, 5, 2, 1, 6, 4, 2, 11, 2, 5, 13, 4, 10, 17, 3, 1, 15, 22, 4, 2, 25, 27, 2, 5, 37, 29, 6, 10, 52, 37, 2, 20, 67, 44, 4, 1, 30, 97, 44, 4, 2, 52, 117, 55, 5, 5, 77, 154, 59, 2, 10, 117, 184, 68, 6, 20, 162, 235, 71, 2, 36, 227, 277, 81, 6, 1, 58, 309, 338
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).
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REFERENCES
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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
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T(n, k) = Partitions of n using only k types of piles. Also, Sum_{k=1, .., A003056(n)} T(n, k)*k = A000070(n). Also, Sum_{k=1, .., A003056(n)} T(n, k)*(k-1) = A058884(n). - Naohiro Nomoto.
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)) = Sum_n T(n, k)*q^n.
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EXAMPLE
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Triangle turned on its side begins:
1.2.2.3.2.4..2..4..3..4..2..6 etc
....1.2.5.6.11.13.17.22.27.29 etc
..........1..2..5.10.15.25.37 etc
......................1..2..5 etc
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CROSSREFS
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Diagonals give A000005, A002133, A002134. Cf. A060043, A060044.
Sequence in context: A003113 A152227 A078660 this_sequence A048896 A130831 A151678
Adjacent sequences: A060174 A060175 A060176 this_sequence A060178 A060179 A060180
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KEYWORD
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nonn,tabf,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Mar 20 2001
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EXTENSIONS
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More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 24 2002
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