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A192621
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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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4
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1, 2, 12, 88, 726, 6456, 60392, 585792, 5838764, 59440250, 615431464, 6460681656, 68607630680, 735682014648, 7954732578032, 86635206695808, 949518438959574, 10464751843723840, 115904823140622164, 1289419736206548408, 14401729960605163272
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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)^2, y=1, z=0.
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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)^(2*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)^(2*n) * Product_{k=1..n} (1+x^(k-1))/((1-x^k*A(x)^2)*(1-x^k)), due to the Heine identity.
(3) A(x)^2 = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x)^2)*(1-x^k), due to the Heine identity.
a(n) ~ c * d^n / n^(3/2), where d = 12.042513458183758627924432194393539477581... and c = 0.323075847195701225672585138139173170517867693... - Vaclav Kotesovec, Oct 04 2023
Radius of convergence r = 0.083039143238027913107320323917684421045... = 1/d and A(r) = 1.624363189835514855585723923742556266289... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 4*r^n/(1 - r^(2*n)*A(r)^4) ) 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 + 2*x + 12*x^2 + 88*x^3 + 726*x^4 + 6456*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 + 2*x*A^2/(1-x) + 2*x^2*A^4*(1+x)/((1-x)*(1-x^2)) + 2*x^3*A^6*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3)) + ...
(2) A = 1 + 2*x*A^2/((1-x*A^2)*(1-x)) + 2*x^3*A^4*(1+x)/((1-x*A^2)*(1-x^2*A^2)*(1-x)*(1-x^2)) + 2*x^6*A^6*(1+x)*(1+x^2)/((1-x*A^2)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x)*(1-x^2)*(1-x^3)) + ...
(3) A^2 = 1 + 4*x*A^2/((1-x*A^2)*(1-x)) + 4*x^2*A^4*(1+x)^2/((1-x*A^2)*(1-x^2*A^2)*(1-x)*(1-x^2)) + 4*x^3*A^6*(1+x)^2*(1+x^2)^2/((1-x*A^2)*(1-x^2*A^2)*(1-x^3*A^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 constant d: *) 1/r /. FindRoot[{QPochhammer[-s^2, r] / QPochhammer[s^2, r] == s*((1 + s^2)/(1 - s^2)), QPolyGamma[0, Log[-s^2]/Log[r], r] - QPolyGamma[0, Log[s^2]/Log[r], r] == (2*(s^2/(s^4 - 1)) - 1/2) * Log[r]}, {r, 1/12}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)
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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^2+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sqrt(1+sum(m=1, n, x^m*A^(2*m)*prod(k=1, m, (1+x^(k-1))^2/((1-x^k*A^2 +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*(m+1)/2)*A^(2*m)*prod(k=1, m, (1+x^(k-1))/((1-x^k*A^2 +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^(2*m)*prod(k=1, m, (1+x^(k-1))/(1-x^k+x*O(x^n)))) ); 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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