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A317665 Expansion of 1/Sum_{k>=0} x^(k^2).

%I A317665 #39 Jul 26 2025 16:54:17
%S A317665 1,-1,1,-1,0,1,-2,3,-3,1,2,-6,10,-11,8,0,-14,29,-39,38,-18,-22,74,
%T A317665 -123,144,-110,6,161,-352,491,-484,251,235,-896,1528,-1825,1452,-191,
%U A317665 -1892,4317,-6164,6243,-3488,-2482,10788,-18957,23140,-19085,3858,22025,-52833,77224,-80198,47899
%N A317665 Expansion of 1/Sum_{k>=0} x^(k^2).
%C A317665 Convolution inverse of A010052.
%H A317665 Seiichi Manyama, <a href="/A317665/b317665.txt">Table of n, a(n) for n = 0..10000</a>
%F A317665 G.f.: 2/(1 + theta_3(q)), where theta_3() is the Jacobi theta function.
%F A317665 a(n) = Sum_{k=0..n} (-1)^k * A337165(n,k).
%F A317665 a(0) = 1; a(n) = -Sum_{k=1..n} A010052(k) * a(n-k). - _Seiichi Manyama_, Mar 19 2022
%e A317665 G.f. = 1 - x + x^2 - x^3 + x^5 - 2*x^6 + 3*x^7 - 3*x^8 + x^9 + 2*x^10 - 6*x^11 + ...
%p A317665 a:=series(1/add(x^(k^2),k=0..100),x=0,54): seq(coeff(a,x,n),n=0..53); # _Paolo P. Lava_, Apr 02 2019
%p A317665 # second Maple program:
%p A317665 a:= proc(n) option remember; `if`(n=0, 1,
%p A317665       -add(`if`(issqr(n-j), a(j), 0), j=0..n-1))
%p A317665     end:
%p A317665 seq(a(n), n=0..53);  # _Alois P. Heinz_, Jul 26 2025
%t A317665 nmax = 53; CoefficientList[Series[1/Sum[x^k^2, {k, 0, nmax}], {x, 0, nmax}], x]
%t A317665 nmax = 53; CoefficientList[Series[2/(1 + QPochhammer[x^2]^5/(QPochhammer[x] QPochhammer[x^4])^2), {x, 0, nmax}], x]
%t A317665 nmax = 53; CoefficientList[Series[2/(1 + EllipticTheta[3, 0, q]), {q, 0, nmax}], q]
%t A317665 a[0] = 1; a[n_] := a[n] = -Sum[Boole[IntegerQ[Sqrt[k]]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 53}]
%o A317665 (PARI) seq(n)=Vec(1/(sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))) \\ _Andrew Howroyd_, Aug 08 2018
%o A317665 (PARI) a(n) = if(n==0, 1, -sum(k=1, n, issquare(k)*a(n-k))); \\ _Seiichi Manyama_, Mar 19 2022
%Y A317665 Cf. A004402, A006456, A010052, A106507, A323633, A337165.
%K A317665 sign
%O A317665 0,7
%A A317665 _Ilya Gutkovskiy_, Aug 08 2018