A192785 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n)*(1 + x^(2*n)*A(x)^2).
1, 1, 2, 5, 11, 29, 75, 203, 557, 1561, 4427, 12706, 36819, 107576, 316579, 937471, 2791487, 8352973, 25104573, 75749240, 229379444, 696851166, 2123304184, 6487295518, 19870096689, 61001089214, 187673207413, 578532522637, 1786712575547
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Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 29*x^5 + 75*x^6 + ... The g.f. A = A(x) satisfies the following relations: A = (1+x)*(1+x^2*A^2)* (1+x^2)*(1+x^4*A^2)* (1+x^3)*(1+x^6*A^2)* ... 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)) + ...
Programs
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Mathematica
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 *) (* 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 *)
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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)} -
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)}
Formula
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.
From Vaclav Kotesovec, Mar 04 2024: (Start)
Let A(x) = y, then 2*y*(1 + y^2) = QPochhammer(-1, x) * QPochhammer(-y^2, x^2).
a(n) ~ c * d^n / n^(3/2), where
d = 3.25215123067662173854186425074452291189580485719079882122325713176...,
c = 1.30862302149248708183423553797270804891358016970005788341511105232...
Radius of convergence:
r = 1/d = 0.307488775604062671485504670197489134974315527740973676344144395...
A(r) = s = 2.80682163771231540175973628784430270489737819467327067575665055...
(End)
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
Comments