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A134337 Number of partitions into distinct odd squarefree parts.

%I A134337 #30 Sep 25 2024 04:12:24
%S A134337 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,
%T A134337 14,14,16,17,20,21,21,25,27,27,29,31,35,35,36,42,44,45,49,55,59,61,66,
%U A134337 74,77,81,87,93,99,102,110,117,123,131,138,148,159,167,178,190,204,215,225
%N A134337 Number of partitions into distinct odd squarefree parts.
%C A134337 Also number of partitions into distinct parts m such that 2*m is squarefree
%H A134337 Vincenzo Librandi and Alois P. Heinz, <a href="/A134337/b134337.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 0..200 from Vincenzo Librandi)
%H A134337 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 16.4.3 "Partitions into square-free parts", pp.351-352
%F A134337 G.f.: Product_{n>=1} (1 + moebius(2*n-1)^2 * x^(2*n-1)).
%F A134337 G.f.: Product_{n>=1} (1 + moebius(2*n)^2 * x^n).
%p A134337 with(numtheory):
%p A134337 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A134337       b(n, i-2)+`if`(i>n or not issqrfree(i), 0, b(n-i, i-2))))
%p A134337     end:
%p A134337 a:= n-> b(n, n-1+irem(n, 2)):
%p A134337 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jul 23 2013
%t A134337 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_ *)
%o A134337 (PARI) my(N=75, x='x+O('x^N)); Vec( prod(n=1,N, 1 + moebius(2*n-1)^2 * x^(2*n-1) ) )
%o A134337 (PARI) my(N=75, x='x+O('x^N)); Vec( prod(n=1,N, 1 + moebius(2*n)^2 * x^n ) )
%K A134337 nonn
%O A134337 0,9
%A A134337 _Joerg Arndt_, Aug 27 2008