A134337 Number of partitions into distinct odd squarefree parts.

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.: 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-François 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: A331904 A025829 A029285 * A261733 A268341 A053633

Adjacent sequences: A134334 A134335 A134336 * A134338 A134339 A134340

Keyword

nonn

Author

Joerg Arndt, Aug 27 2008

Status

approved