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A284466
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Number of compositions (ordered partitions) of n into odd divisors of n.
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2
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1, 1, 1, 2, 1, 2, 6, 2, 1, 20, 8, 2, 60, 2, 10, 450, 1, 2, 726, 2, 140, 3321, 14, 2, 5896, 572, 16, 26426, 264, 2, 394406, 2, 1, 226020, 20, 51886, 961584, 2, 22, 2044895, 38740, 2, 20959503, 2, 676, 478164163, 26, 2, 56849086, 31201, 652968, 184947044, 980, 2, 1273706934, 6620376, 153366, 1803937344
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) = [x^n] 1/(1 - Sum_{d|n, d positive odd} x^d).
a(n) = 1 if n is a power of 2.
a(n) = 2 if n is an odd prime.
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EXAMPLE
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a(10) = 8 because 10 has 4 divisors {1, 2, 5, 10} among which 2 are odd {1, 5} therefore we have [5, 5], [5, 1, 1, 1, 1, 1], [1, 5, 1, 1, 1, 1], [1, 1, 5, 1, 1, 1], [1, 1, 1, 5, 1, 1], [1, 1, 1, 1, 5, 1], [1, 1, 1, 1, 1, 5] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
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MAPLE
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with(numtheory):
a:= proc(n) option remember; local b, l;
l, b:= select(x-> is(x:: odd), divisors(n)),
proc(m) option remember; `if`(m=0, 1,
add(`if`(j>m, 0, b(m-j)), j=l))
end; b(n)
end:
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MATHEMATICA
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Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[Mod[d[[k]], 2] == 1] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 57}]
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PROG
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(Python)
from sympy import divisors
from sympy.core.cache import cacheit
@cacheit
def a(n):
l=[x for x in divisors(n) if x%2]
@cacheit
def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
return b(n)
print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 01 2017, after Maple code
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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