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%I #12 Sep 07 2023 12:47:40
%S 1,0,2,2,3,4,5,8,9,12,15,20,23,28,36,44,52,62,76,90,106,124,149,176,
%T 203,236,279,324,372,430,499,576,657,752,867,992,1124,1280,1463,1662,
%U 1876,2124,2410,2722,3061,3446,3889,4374,4896,5490,6166,6900,7700,8600
%N Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.
%H G. C. Greubel, <a href="/A338463/b338463.txt">Table of n, a(n) for n = 2..1002</a>
%F G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^2.
%F a(n) = Sum_{k=1..n-1} A000700(k) * A000700(n-k).
%F a(n) = A073252(n) - 2 * A000700(n) for n > 0.
%F a(n) = [x^n]( (2/QPochhammer(-1,-x) - 1)^2 ). - _G. C. Greubel_, Sep 07 2023
%t nmax = 55; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
%t With[{k=2}, Drop[CoefficientList[Series[(2/QPochhammer[-1,-x] -1)^k, {x,0,80}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)
%o (Magma)
%o m:=80;
%o R<x>:=PowerSeriesRing(Integers(), m);
%o Coefficients(R!( (-1 + (&*[1+x^(2*j+1): j in [0..m+2]]) )^2 )); // _G. C. Greubel_, Sep 07 2023
%o (SageMath)
%o m=80
%o def f(x): return (-1 + product(1+x^(2*j-1) for j in range(1,m+3)) )^2
%o def A338463_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( f(x) ).list()
%o a=A338463_list(m); a[2:] # _G. C. Greubel_, Sep 07 2023
%Y Cf. A000700, A022597, A047654, A048574, A073252, A327380, A338223.
%K nonn
%O 2,3
%A _Ilya Gutkovskiy_, Jan 31 2021
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