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A192620
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G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/(1 - x^n*A(x))^2.
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5
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1, 4, 28, 224, 1948, 17928, 171776, 1695872, 17133436, 176297668, 1841222776, 19467629120, 207978652416, 2241618514120, 24345336854400, 266168049520832, 2927074607294300, 32356419163487336, 359330087240388828, 4007079691584624576
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OFFSET
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0,2
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COMMENTS
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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=z=1.
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = 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.
(2) A(x)^(1/2) = 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.
Equals the self-convolution of A192621.
a(n) ~ c * d^n / n^(3/2), where d = 12.042513458183758627924432194393539477581... and c = 1.04958502741924123967536156787764354342367951743839... - Vaclav Kotesovec, Oct 04 2023
Radius of convergence r = 0.083039143238027913107320323917684421045... = 1/d and A(r) = 2.638555772492608872250287025192536127217... satisfy A(r) = 1 / Sum_{n>=1} 4*r^n/(1 - r^(2*n)*A(r)^2) and A(r) = Product_{n>=1} (1 + r^n*A(r))^2/(1 - r^n*A(r))^2. - Paul D. Hanna, Mar 02 2024
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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 28*x^2 + 224*x^3 + 1948*x^4 + 17928*x^5 + ...
The g.f. A = A(x) satisfies the following relations:
(0) A = (1+x*A)^2/(1-x*A)^2 * (1+x^2*A)^2/(1-x^2*A)^2 * (1+x^3*A)^2/(1-x^3*A)^2 * ...
(1) A = 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)) + ...
(2) A^(1/2) = 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)) + ...
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MATHEMATICA
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nmax = 30; A[_] = 0; Do[A[x_] = Product[(1 + x^k*A[x])^2/(1 - x^k*A[x])^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 04 2023 *)
(* Calculation of constants {d, c}: *) Chop[{1/r, Sqrt[(r*s^(3/2)*((-1 + s)*Derivative[0, 1][QPochhammer][-s, r] + Sqrt[s]*(1 + s)*Derivative[0, 1][QPochhammer][s, r]))/(2* Pi*(1 + s)*QPochhammer[s, r]* (2* s*((1 + s^2)/(-1 + s^2)^2) + (QPolyGamma[1, Log[-s]/Log[r], r] - QPolyGamma[1, Log[s]/Log[r], r])/ Log[r]^2))]} /. FindRoot[{(-1 + s)^2*(QPochhammer[-s, r]^2/((1 + s)^2*QPochhammer[s, r]^2)) == s, 1 - 4*(s/(-1 + s^2)) + (2*(QPolyGamma[0, Log[-s]/Log[r], r] - QPolyGamma[0, Log[s]/Log[r], r]))/Log[r] == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Mar 03 2024 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(k=1, n, (1+x^k*A)^2/(1-x^k*A+x*O(x^n))^2)); 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))^2/((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=(1+sum(m=1, n, x^m*A^m*prod(k=1, m, (1+x^(k-1))/(1-x^k+x*O(x^n)))))^2); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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