0,2

Related q-series (Heine) identity:

1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)); here q=x, x=x*A(x), y=1, z=0.

G.f. satisfies:

(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^n*Product_{k=1..n} (1+x^(k-1))/(1-x^k) due to the q-binomial theorem.

(2) A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*A(x)^n*Product_{k=1..n} (1+x^(k-1))/((1-x^k*A(x))*(1-x^k)) due to the Heine identity.

(3) A(x)^2 = 1 + Sum_{n>=1} x^n*A(x)^n * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x))*(1-x^k)) due to the Heine identity.

a(n) ~ c * d^n / n^(3/2), where d = 6.6934289011143535333002543297069340451347... and c = 0.946606599119645056034760125205426820822370610602636232678... - Vaclav Kotesovec, Sep 26 2023

Radius of convergence r = 0.149400257293166331446262618504038357688... = 1/d and A(r) = 2.500666835731534833961673247439001530869... satisfy A(r) = 1 / Sum_{n>=1} 2*r^n/(1 - r^(2*n)*A(r)^2) and A(r) = Product_{n>=1} (1 + r^n*A(r))/(1 - r^n*A(r)). - Paul D. Hanna, Mar 02 2024

G.f.: A(x) = 1 + 2*x + 8*x^2 + 36*x^3 + 174*x^4 + 888*x^5 + ...

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

(0) A = (1+x*A)/(1-x*A) * (1+x^2*A)/(1-x^2*A) * (1+x^3*A)/(1-x^3*A) * ...

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

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

(3) A^2 = 1 + 4*x*A/((1-x*A)*(1-x)) + 4*x^2*A^2*(1+x)^2/((1-x*A)*(1-x^2*A)*(1-x)*(1-x^2)) + 4*x^3*A^3*(1+x)^2*(1+x^2)^2/((1-x*A)*(1-x^2*A)*(1-x^3*A)*(1-x)*(1-x^2)*(1-x^3)) + ... (cf. A192619)

nmax = 30; A[_] = 0; Do[A[x_] = Product[(1 + x^k*A[x])/(1 - x^k*A[x]), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)

(* Calculation of constant d: *) 1/r /. FindRoot[{(s-1)*QPochhammer[-s, r] == -s*(s+1) * QPochhammer[s, r], (s^2 - 1)*(QPolyGamma[0, Log[-s]/Log[r], r] - QPolyGamma[0, Log[s]/Log[r], r]) + Log[r]*(s^2 - 2*s - 1) == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 26 2023 *)

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+x^m*A)/(1-x^m*A+x*O(x^n)))); polcoeff(A, n)}

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

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

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

Cf. A145267, A145268, A190861, A192619 (g.f. A(x)^2), A192621.

Sequence in context: A352862 A109980 A186338 * A380046 A110837 A372088

Adjacent sequences: A190859 A190860 A190861 * A190863 A190864 A190865

Paul D. Hanna, May 21 2011

approved