cp's OEIS Frontend

A100522 Number of partitions of n into parts free of both odd squares and even numbers which are not squares, the odd parts they occur with a single multiplicity, there is no restriction on the even parts.

%I A100522 #15 Mar 31 2023 05:13:21
%S A100522 1,0,0,1,1,1,0,2,2,1,1,3,3,2,2,5,6,3,5,8,9,7,8,13,14,10,14,19,20,17,
%T A100522 20,29,30,26,32,42,45,41,47,63,64,60,70,88,91,87,99,124,128,123,143,
%U A100522 172,179,176,200,240,246,246,279,325,337,338,381,440,456,461,519,590,615
%N A100522 Number of partitions of n into parts free of both odd squares and even numbers which are not squares, the odd parts they occur with a single multiplicity, there is no restriction on the even parts.
%H A100522 G. C. Greubel, <a href="/A100522/b100522.txt">Table of n, a(n) for n = 0..5000</a>
%H A100522 Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011, 2004.
%F A100522 G.f.: Product_{k>0} (1+x^(2*k-1))/(1-(-1)^k*x^(k^2)).
%e A100522 a(16)=6 because 16 = 13+3 = 11+5 = 7+5+4 = 5+3+4+4 = 4+4+4+4.
%p A100522 series(product((1+x^(2*k-))/(1-(-1)^k*x^(k^2)),k=1..100),x=0,100);
%t A100522 With[{m=80}, CoefficientList[Series[Product[(1+x^(2*k-1))/(1-(-1)^k *x^(k^2)), {k,m+2}], {x,0,m}], x]] (* _G. C. Greubel_, Mar 28 2023 *)
%o A100522 (Magma)
%o A100522 m:=80;
%o A100522 f:= func< x | (&*[(1+x^(2*k-1))/(1-(-1)^k*x^(k^2)): k in [1..m+2]]) >;
%o A100522 R<x>:=PowerSeriesRing(Integers(), m);
%o A100522 Coefficients(R!( f(x) )); // _G. C. Greubel_, Mar 28 2023
%o A100522 (SageMath)
%o A100522 m=80
%o A100522 def f(x): return product( (1+x^(2*k-1))/(1-(-1)^k*x^(k^2)) for k in range(1,m+2))
%o A100522 def A100522_list(prec):
%o A100522     P.<x> = PowerSeriesRing(ZZ, prec)
%o A100522     return P( f(x) ).list()
%o A100522 A100522_list(m) # _G. C. Greubel_, Mar 28 2023
%Y A100522 Cf. A100926.
%K A100522 nonn
%O A100522 0,8
%A A100522 _Noureddine Chair_, Nov 25 2004
%E A100522 Offset corrected by _G. C. Greubel_, Mar 28 2023