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Number of partitions of n into odd squares.
11

%I #19 Sep 18 2017 17:54:36

%S 1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,

%T 6,6,7,7,7,7,7,7,7,8,8,9,9,9,9,10,11,11,12,12,13,13,13,13,14,15,15,16,

%U 16,17,17,17,17,18,19,19,20,20,21,21,22,23,24,25,25,26,26,28,28,29,30,31

%N Number of partitions of n into odd squares.

%C A167662 and A167663 give record values and where they occur: A167662(n)=a(A167663(n)) and a(m) < A167662(n) for m < A167663(n).

%H Vaclav Kotesovec, <a href="/A167661/b167661.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from R. Zumkeller)

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>.

%F a(n) = f(n,1,8) with f(x,y,z) = if x<y then 0^x else f(x-y,y,z)+f(x,y+z,z+8).

%F G.f.: G = 1/Product_{i>=1}(1-x^{(2i-1)^2}). - _Emeric Deutsch_ , Jan 26 2016

%F a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 4) * Zeta(3/2)^(1/3) / (4 * sqrt(3) * Pi^(1/3) * n^(5/6)). - _Vaclav Kotesovec_, Sep 18 2017

%e a(10)=#{9+1,1+1+1+1+1+1+1+1+1+1}=2;

%e a(20)=#{9+9+1+1,9+1+1+1+1+1+1+1+1+1+1+1,20x1}=3;

%e a(30)=#{25+1+1+1+1+1,9+9+9+1+1+1,9+9+12x1,9+21x1,30x1}=5.

%p g := 1/mul(1-x^((2*i-1)^2), i = 1 .. 150): gser := series(g, x = 0, 105): seq(coeff(gser, x, n), n = 0 .. 100);

%t nmax = 100; CoefficientList[Series[Product[1/(1 - x^((2*k-1)^2)), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 18 2017 *)

%Y Cf. A001156, A000009, A101412, A016754, A167700.

%K nonn

%O 0,10

%A _Reinhard Zumkeller_, Nov 08 2009