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A134337
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Number of partitions into distinct odd squarefree parts.
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4
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1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 3, 4, 3, 4, 5, 5, 6, 6, 7, 8, 7, 8, 9, 9, 11, 10, 12, 14, 14, 16, 17, 20, 21, 21, 25, 27, 27, 29, 31, 35, 35, 36, 42, 44, 45, 49, 55, 59, 61, 66, 74, 77, 81, 87, 93, 99, 102, 110, 117, 123, 131, 138, 148, 159, 167, 178, 190, 204, 215, 225
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OFFSET
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0,9
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COMMENTS
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Also number of partitions into distinct parts m such that 2*m is squarefree
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LINKS
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FORMULA
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G.f.: prod(n>=1, 1 + moebius(2*n-1)^2 * x^(2*n-1) ) ) = prod(n>=1, 1 + moebius(2*n)^2 * x^(n) ) )
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MAPLE
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with(numtheory):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-2)+`if`(i>n or not issqrfree(i), 0, b(n-i, i-2))))
end:
a:= n-> b(n, n-1+irem(n, 2)):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2] + If[i>n || !SquareFreeQ[i] , 0, b[n-i, i-2]]]]; a[n_] := b[n, n-1 + Mod[n, 2]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 08 2015, after Alois P. Heinz *)
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PROG
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(PARI) N=75; x='x+O('x^N); Vec( prod(n=1, N, 1 + moebius(2*n-1)^2 * x^(2*n-1) ) )
(PARI) N=75; x='x+O('x^N); Vec( prod(n=1, 100, 1 + moebius(2*n)^2 * x^(n) ) )
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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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