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%I #25 Mar 03 2024 09:57:00
%S 1,1,4,18,92,505,2922,17541,108270,682823,4380942,28504466,187636994,
%T 1247375147,8362420498,56471709841,383790966537,2622982116829,
%U 18016055333571,124296340608870,860986586024343,5985590694574930,41749023026002831
%N G.f. satisfies A(x) = 1/Product_{n>=1} (1 - x^n*A(x)^2).
%H Vaclav Kotesovec, <a href="/A196150/b196150.txt">Table of n, a(n) for n = 0..150</a>
%F G.f. satisfies:
%F (1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) / Product_{k=1..n} (1-x^k) due to an identity of Euler.
%F (2) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^(2*n)/[Product_{k=1..n} (1-x^k)*(1-x^k*A(x)^2)] due to Cauchy's identity.
%F (3) A(x) = 1 + Sum_{n>=1} x^n*A(x)^2 / Product_{k=1..n} (1 - x^k*A(x)^2).
%F a(n) ~ c * d^n / n^(3/2), where d = 7.4702934491577480082... and c = 0.270144986991156076... - _Vaclav Kotesovec_, Aug 12 2021
%F Radius of convergence r = 0.1338635499135240586... = 1/d and A(r) = 1.5228379370493260575... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 2*r^n/(1 - r^n*A(r)^2) ) and A(r) = 1 / Product_{n>=1} (1 - r^n*A(r)^2). - _Paul D. Hanna_, Mar 02 2024
%e G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 92*x^4 + 505*x^5 + 2922*x^6 + ...
%e where
%e (0) A(x) = 1/((1-x*A(x)^2) * (1-x^2*A(x)^2) * (1-x^3*A(x)^2) * ...).
%e (1) A(x) = 1 + x*A(x)^2/(1-x) + x^2*A(x)^4/((1-x)*(1-x^2)) + x^3*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + ...
%e (2) A(x) = 1 + x*A(x)^2/[(1-x)*(1-x*A(x)^2)] + x^4*A(x)^4/[(1-x)*(1-x^2)*(1-x*A(x)^2)*(1-x^2*A(x)^2)] + x^9*A(x)^6/[(1-x)*(1-x^2)*(1-x^3)*(1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)] + ...
%e (3) A(x) = 1 + x*A(x)^2/(1-x*A(x)^2) + x^2*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)) + x^3*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)) + ...
%t nmax = 30; A[_] = 0; Do[A[x_] = 1/Product[1 - x^k*A[x]^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* _Vaclav Kotesovec_, Sep 28 2023 *)
%t (* Calculation of constants {d,c}: *) {1/r, Sqrt[r*s^3 * (s^2 - 1) * Derivative[0, 1][QPochhammer][s^2, r] / (8*Pi*((s^2 - 1)^2*(QPolyGamma[1, 2*Log[s]/Log[r], r] / Log[r]^2) - s^2))]} /. FindRoot[{(1 - s^2)/QPochhammer[s^2, r] == s, 1/2 + s^2/(1 - s^2) == (Log[1 - r] + QPolyGamma[0, 2*Log[s]/Log[r], r]) / Log[r]}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Sep 28 2023 *)
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*A^2+x*O(x^n)))); polcoeff(A, n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(2*m)/prod(k=1, m, (1-x^k+x*O(x^n))))); polcoeff(A, n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(2*m)/prod(k=1, m, (1-x^k)*(1-x^k*A^2+x*O(x^n))))); polcoeff(A, n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^2/prod(k=1, m, (1-x^k*A^2+x*O(x^n))))); polcoeff(A, n)}
%Y Cf. A196151, A145268, A145267, A206639, A206637.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 28 2011
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