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A167700 Number of partitions of n into distinct odd squares.

%I A167700 #26 Jan 09 2023 12:33:02
%S A167700 1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,
%T A167700 1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,
%U A167700 0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A167700 Number of partitions of n into distinct odd squares.
%C A167700 A167701 and A167702 give record values and where they occur: A167701(n)=a(A167702(n)) and a(m) < A167701(n) for m < A167702(n);
%C A167700 a(A167703(n)) = 0.
%H A167700 Reinhard Zumkeller, <a href="/A167700/b167700.txt">Table of n, a(n) for n = 0..10000</a>
%H A167700 Vaclav Kotesovec, <a href="/A167700/a167700.jpg">Graph - The asymptotic ratio</a>
%H A167700 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>.
%F A167700 a(n) = f(n,1,8) with f(x,y,z) = if x<y then 0^x else f(x-y,y+z,z+8) + f(x,y+z,z+8).
%F A167700 G.f.: Product_{k>=0} (1 + x^((2*k+1)^2)). - _Ilya Gutkovskiy_, Jan 11 2017
%F A167700 a(n) ~ exp(3 * 2^(-7/3) * Pi^(1/3) * (sqrt(2)-1)^(2/3) * Zeta(3/2)^(2/3) * n^(1/3)) * (sqrt(2)-1)^(1/3) * Zeta(3/2)^(1/3) / (2^(7/6) * sqrt(3) * Pi^(1/3) * n^(5/6)). - _Vaclav Kotesovec_, Sep 18 2017
%e A167700 a(50) = #{49+1} = 1;
%e A167700 a(130) = #{121+9, 81+49} = 2.
%t A167700 nmax = 100; CoefficientList[Series[Product[1 + x^((2*k-1)^2), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 18 2017 *)
%o A167700 (Haskell)
%o A167700 a167700 = p a016754_list where
%o A167700    p _  0 = 1
%o A167700    p (q:qs) m = if m < q then 0 else p qs (m - q) + p qs m
%o A167700 -- _Reinhard Zumkeller_, Mar 15 2014
%Y A167700 Cf. A033461, A016754, A167661, A000700, A111900.
%K A167700 nonn,look
%O A167700 0,131
%A A167700 _Reinhard Zumkeller_, Nov 09 2009