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A134337 Number of partitions into distinct odd squarefree parts. 2
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 (list; graph; refs; listen; history; text; internal format)
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.: prod(n>=1, 1 + moebius(2*n-1)^2 * x^(2*n-1) ) ) =  prod(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-Fran├žois Alcover, Apr 08 2015, after Alois P. Heinz *)

PROG

(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) ) )

CROSSREFS

Sequence in context: A190353 A025829 A029285 * A261733 A268341 A053633

Adjacent sequences:  A134334 A134335 A134336 * A134338 A134339 A134340

KEYWORD

nonn

AUTHOR

Joerg Arndt, Aug 27 2008

STATUS

approved

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Last modified March 22 17:25 EDT 2019. Contains 321422 sequences. (Running on oeis4.)