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

Author

Joerg Arndt, Aug 27 2008

Keywords

• nonn

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

Programs

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 *)

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 ) )

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).