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A134337
Number of partitions into distinct odd squarefree parts.
4
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
OFFSET
0,9
COMMENTS
Also number of partitions into distinct parts m such that 2*m is squarefree
LINKS
Vincenzo Librandi and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Vincenzo Librandi)
Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.3 "Partitions into square-free parts", pp.351-352
FORMULA
G.f.: Product_{n>=1} (1 + moebius(2*n-1)^2 * x^(2*n-1)).
G.f.: Product_{n>=1} (1 + moebius(2*n)^2 * x^n).
MAPLE
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)):
seq(a(n), n=0..100); # Alois P. Heinz, Jul 23 2013
MATHEMATICA
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 *)
PROG
(PARI) my(N=75, x='x+O('x^N)); Vec( prod(n=1, N, 1 + moebius(2*n-1)^2 * x^(2*n-1) ) )
(PARI) my(N=75, x='x+O('x^N)); Vec( prod(n=1, N, 1 + moebius(2*n)^2 * x^n ) )
CROSSREFS
Sequence in context: A331904 A025829 A029285 * A261733 A268341 A053633
KEYWORD
nonn
AUTHOR
Joerg Arndt, Aug 27 2008
STATUS
approved