A192785 - OEIS

%I #16 Mar 07 2024 02:28:34

%S 1,1,2,5,11,29,75,203,557,1561,4427,12706,36819,107576,316579,937471,

%T 2791487,8352973,25104573,75749240,229379444,696851166,2123304184,

%U 6487295518,19870096689,61001089214,187673207413,578532522637,1786712575547

%N  G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n)*(1 + x^(2*n)*A(x)^2).

%C Related q-series identity of Lebesgue:

%C Product_{n>=1} (1 + q^n)*(1 + z*q^(2*n)) = 1 + Sum_{n>=1} q^(n*(n+1)/2) * Product_{k=1..n} (1 + z*q^k)/(1 - x^k); here q=x, z=A(x)^2.

%F G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*Product_{k=1..n} (1 + x^k*A(x)^2)/(1 - x^k) due to a Lebesgue identity.

%F From _Vaclav Kotesovec_, Mar 04 2024: (Start)

%F Let A(x) = y, then 2*y*(1 + y^2) = QPochhammer(-1, x) * QPochhammer(-y^2, x^2).

%F a(n) ~ c * d^n / n^(3/2), where

%F d = 3.25215123067662173854186425074452291189580485719079882122325713176...,

%F c = 1.30862302149248708183423553797270804891358016970005788341511105232...

%F Radius of convergence:

%F r = 1/d = 0.307488775604062671485504670197489134974315527740973676344144395...

%F A(r) = s = 2.80682163771231540175973628784430270489737819467327067575665055...

%F (End)

%F The values r and A(r) given above also satisfy A(r) = 1 / sqrt( Sum_{n>=1} 2*r^(2*n)/(1 + r^(2*n)*A(r)^2) ). - _Paul D. Hanna_, Mar 06 2024

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 29*x^5 + 75*x^6 + ...

%e The g.f. A = A(x) satisfies the following relations:

%e A = (1+x)*(1+x^2*A^2)* (1+x^2)*(1+x^4*A^2)* (1+x^3)*(1+x^6*A^2)* ...

%e A = 1 + x*(1+x*A^2)/(1-x) + x^3*(1+x*A^2)*(1+x^2*A^2)/((1-x)*(1-x^2)) + x^6*(1+x*A^2)*(1+x^2*A^2)*(1+x^3*A^2)/((1-x)*(1-x^2)*(1-x^3)) + ...

%t nmax = 30; A[_] = 0; Do[A[x_] = Product[(1 + x^k)*(1 + x^(2*k)*A[x]^2), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* _Vaclav Kotesovec_, Mar 04 2024 *)

%t (* Calculation of constants {d,c}: *) Chop[{1/r, Sqrt[(-r*s*(1 + s^2) * Log[r]^2 * ((s + s^3)*Derivative[0, 1][QPochhammer][-1, r] + r*QPochhammer[-1, r]^2 * Derivative[0, 1][QPochhammer][-s^2, r^2]))/ (2*Pi * QPochhammer[-1, r]*(4*s^2*Log[r]^2 + (1 + s^2)^2 * QPolyGamma[1, Log[-s^2]/(2*Log[r]), r^2]))]} /. FindRoot[{(QPochhammer[-1, r]*QPochhammer[-s^2, r^2])/(2 + 2*s^2) == s, 1 + 3*s^2 + (1 + s^2)*((Log[1 - r^2] + QPolyGamma[0, Log[-s^2]/(2*Log[r]), r^2])/Log[r]) == 0}, {r, 1/3}, {s, 2}, WorkingPrecision -> 70]] (* _Vaclav Kotesovec_, Mar 04 2024 *)

%o (PARI) {a(n) = my(A=1+x); for(i=1, n, A = prod(m=1, n, (1 + x^m)*(1 + x^(2*m)*A^2 +x*O(x^n)))); polcoeff(A, n)}

%o (PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + sum(m=1, sqrtint(2*n), x^(m*(m+1)/2)*prod(k=1, m, (1 + A^2*x^k)/(1 - x^k+x*O(x^n))))); polcoeff(A, n)}

%Y Cf. A190822, A145267.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 10 2011