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A060177
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Triangle of generalized sum of divisors function, read by rows.
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15
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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
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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 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).
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LINKS
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FORMULA
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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 ...
1 2 5 6 11 13 17 22 27 29 ...
1 2 5 10 15 25 37 ...
1 2 5 ...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
expand(b(n, i-1) +x*add(b(n-i*j, i-1), j=1..n/i))))
end:
T:= n->(p->seq(coeff(p, x, degree(p)-k), k=0..degree(p)-1))(b(n$2)):
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MATHEMATICA
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Reverse /@ Table[Length /@ Split[ Sort[Map[Length, Split /@ IntegerPartitions[n], {1}]]], {n, 24}] (* Wouter Meeussen, Apr 21 2012, updated by Jean-François Alcover, Jan 29 2014 *)
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PROG
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(Python)
from math import isqrt
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A060177_gen(): # generator of terms
return (sum(1 for p in partitions(n) if len(p)==k) for n in count(1) for k in range(isqrt((n<<3)+1)-1>>1, 0, -1))
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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