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%I A166952 #15 Nov 16 2023 11:25:31
%S A166952 1,2,4,8,18,52,184,688,2512,8866,30824,108088,387952,1426804,5335152,
%T A166952 20105808,75979458,287627524,1092023532,4163964648,15955084784,
%U A166952 61412039424,237256107576,919294150288,3570699037984,13900290723814
%N A166952 G.f. satisfies: A(x) = theta_3(x*A(x)) where Jacobi theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%C A166952 From _Paul D. Hanna_, Apr 24 2010: (Start)
%C A166952 SPECIAL VALUES:
%C A166952 . at x = exp(-Pi)*Pi^(-1/4)*gamma(3/4) = 0.039775896186087627425...,
%C A166952 . A(x) = theta_3(exp(-Pi)) = Pi^(1/4)/gamma(3/4) = 1.0864348112133080...
%C A166952 RADIUS OF CONVERGENCE r:
%C A166952 . at r = 0.241970723224463308846762732757915397312...,
%C A166952 . A(r) = 2.506628552782237708927560606516272396709...
%C A166952 where r and A(r) are given by:
%C A166952 . r = z/theta_3(z) and
%C A166952 . A(r) = theta_3(z)
%C A166952 such that z is the real root nearest the origin that satisfies:
%C A166952 . theta_3(z) - z*theta_3'(z) = 0, which has solution:
%C A166952 . z = 0.6065307237718078589943387177361885081872...
%C A166952 (End)
%F A166952 G.f.: A(x) = (1/x)*Series_Reversion(x/theta_3(x)).
%F A166952 G.f. satisfies: A(x/theta_3(x)) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%F A166952 G.f. satisfies: A(x) = Product_{n>=1} (1 - (-x)^n*A(x)^n) / (1 + (-x)^n*A(x)^n).
%F A166952 G.f. satisfies: A(x) = Product_{n>=1} (1 + (x*A(x))^(2*n-1))^2 * (1 - (x*A(x))^(2*n)).
%F A166952 G.f. satisfies: [x^n] A(x)^(1-n) = 2-2n if n>0 is square, zero otherwise.
%F A166952 a(n) = A066536(n)/(n+1) where A066536(n) equals the number of ways of writing n as the sum of n+1 squares.
%F A166952 Logarithmic derivative yields A066535, number of ways of writing n as the sum of n squares, for n>=1.
%F A166952 a(n) ~ c * d^n / n^(3/2), where d = 4.13273137623493996302796465513832835490078625705045019249993320055571... and c = 0.70710538549959357505200420443014251744770906948354300807129911827348... - _Vaclav Kotesovec_, Nov 16 2023
%e A166952 G.f.: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 18*x^4 + 52*x^5 + 184*x^6 +...
%e A166952 A(x/theta_3(x)) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 +...
%e A166952 log(A(x)) = 2*x + 4*x^2/2 + 8*x^3/3 + 24*x^4/4 + 112*x^5/5 +...+ A066535(n)*x^n/n +...
%t A166952 CoefficientList[1/x*InverseSeries[Series[x/EllipticTheta[3, 0, x], {x, 0, 25}], x], x] (* _Vaclav Kotesovec_, Nov 16 2023 *)
%t A166952 (* Calculation of constants {d,c}: *) {1/r, Sqrt[s/(2*Pi*r^2*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s], r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/4}, {s, 5/2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Nov 16 2023 *)
%o A166952 (PARI) a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n));polcoeff(THETA3^(n+1),n)/(n+1)
%o A166952 (PARI) a(n)=local(A=1+x); for(i=1, n, A=prod(k=1,n,(1-(-x)^k*A^k+x*O(x^n))/(1+(-x)^k*A^k+x*O(x^n)) )); polcoeff(A, n)
%o A166952 for(n=0,30,print1(a(n),", "))
%o A166952 (PARI) a(n)=local(A=1+x); for(i=1, n, A=prod(k=1,n,(1+(x*A)^(2*k-1)+x*O(x^n))^2*(1-(x*A)^(2*k)+x*O(x^n)) )); polcoeff(A, n)
%o A166952 for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Jul 12 2013
%Y A166952 Cf. A000122 (theta_3), A066535, A066536.
%K A166952 nonn
%O A166952 0,2
%A A166952 _Paul D. Hanna_, Oct 26 2009

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