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A060047
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Triangle of generalized sum of divisors function, read by rows.
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4
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1, 2, 4, 1, 4, 2, 6, 4, 8, 8, 8, 14, 8, 1, 18, 13, 2, 28, 12, 4, 40, 12, 8, 52, 16, 14, 70, 14, 24, 88, 16, 40, 104, 24, 1, 56, 140, 16, 2, 84, 168, 18, 4, 122, 196, 26, 8, 168, 240, 20, 14, 232, 278, 24, 24, 312, 320, 32, 40, 408, 380, 24, 64, 528, 440, 24, 100, 672, 504
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OFFSET
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1,2
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COMMENTS
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Lengths of rows are 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 ... (A000196).
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LINKS
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Table of n, a(n) for n=1..69.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
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FORMULA
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T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*(2*m_1-1) + s_2*(2*m_2-1) + ... + s_k*(2*m_k-1) = 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^(2*m_1+2*m_2+...+2*m_k-k)/((1-q^{2*m_1-1})*(1-q^{2*m_2-1})*...*(1-q^{2*m_k-1}))^2 = Sum_n T(n, k)*q^n.
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EXAMPLE
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Triangle turned on its side begins:
1 2 4 4 6 8 8 8 13 12 12 ...
1 2 4 8 14 18 28 40 ...
1 2 4 ...
For example, T(6,1) = 8, T(6,2) = 4.
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CROSSREFS
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Diagonals give A002131, A002132, A060046. Cf. A060043, A060044, A060177, A060184.
Sequence in context: A011167 A014176 A317690 * A135185 A289460 A201774
Adjacent sequences: A060044 A060045 A060046 * A060048 A060049 A060050
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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, Mar 19 2001
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EXTENSIONS
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More terms from Naohiro Nomoto, Jan 24 2002
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STATUS
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approved
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