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A196150
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G.f. satisfies A(x) = 1/Product_{n>=1} (1 - x^n*A(x)^2).
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7
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1, 1, 4, 18, 92, 505, 2922, 17541, 108270, 682823, 4380942, 28504466, 187636994, 1247375147, 8362420498, 56471709841, 383790966537, 2622982116829, 18016055333571, 124296340608870, 860986586024343, 5985590694574930, 41749023026002831
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) / Product_{k=1..n} (1-x^k) due to an identity of Euler.
(2) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^(2*n)/[Product_{k=1..n} (1-x^k)*(1-x^k*A(x)^2)] due to Cauchy's identity.
(3) A(x) = 1 + Sum_{n>=1} x^n*A(x)^2 / Product_{k=1..n} (1 - x^k*A(x)^2).
a(n) ~ c * d^n / n^(3/2), where d = 7.4702934491577480082... and c = 0.270144986991156076... - Vaclav Kotesovec, Aug 12 2021
Radius of convergence r = 0.1338635499135240586... = 1/d and A(r) = 1.5228379370493260575... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 2*r^n/(1 - r^n*A(r)^2) ) and A(r) = 1 / Product_{n>=1} (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 + x + 4*x^2 + 18*x^3 + 92*x^4 + 505*x^5 + 2922*x^6 + ...
where
(0) A(x) = 1/((1-x*A(x)^2) * (1-x^2*A(x)^2) * (1-x^3*A(x)^2) * ...).
(1) A(x) = 1 + x*A(x)^2/(1-x) + x^2*A(x)^4/((1-x)*(1-x^2)) + x^3*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + ...
(2) A(x) = 1 + x*A(x)^2/[(1-x)*(1-x*A(x)^2)] + x^4*A(x)^4/[(1-x)*(1-x^2)*(1-x*A(x)^2)*(1-x^2*A(x)^2)] + x^9*A(x)^6/[(1-x)*(1-x^2)*(1-x^3)*(1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)] + ...
(3) A(x) = 1 + x*A(x)^2/(1-x*A(x)^2) + x^2*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)) + x^3*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)) + ...
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MATHEMATICA
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nmax = 30; A[_] = 0; Do[A[x_] = 1/Product[1 - x^k*A[x]^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 28 2023 *)
(* Calculation of constants {d, c}: *) {1/r, Sqrt[r*s^3 * (s^2 - 1) * Derivative[0, 1][QPochhammer][s^2, r] / (8*Pi*((s^2 - 1)^2*(QPolyGamma[1, 2*Log[s]/Log[r], r] / Log[r]^2) - s^2))]} /. FindRoot[{(1 - s^2)/QPochhammer[s^2, r] == s, 1/2 + s^2/(1 - s^2) == (Log[1 - r] + QPolyGamma[0, 2*Log[s]/Log[r], r]) / Log[r]}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 28 2023 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*A^2+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^(2*m)/prod(k=1, m, (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, sqrtint(n+1), x^(m^2)*A^(2*m)/prod(k=1, m, (1-x^k)*(1-x^k*A^2+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^2/prod(k=1, m, (1-x^k*A^2+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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