A190862 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))/(1 - x^n*A(x)).
1, 2, 8, 36, 174, 888, 4716, 25808, 144568, 825030, 4780176, 28045860, 166295716, 994959560, 5999349896, 36420226288, 222415222446, 1365445230212, 8422174103796, 52168047039764, 324366739546304, 2023789526326096
Offset: 0
Keywords
Examples
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)
Programs
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Mathematica
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 *)
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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)}
Formula
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
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