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A196151
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G.f. satisfies A(x) = Product_{n>=1} (1 + x^n*A(x)^2).
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3
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1, 1, 3, 11, 43, 179, 778, 3491, 16051, 75235, 358170, 1727124, 8418266, 41408344, 205289265, 1024737905, 5145933602, 25978844478, 131773584768, 671239285119, 3432304205872, 17611565623950, 90652384728648, 467963720803022, 2422110238147351
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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) = Sum_{n>=0} x^(n*(n+1)/2)*A(x)^(2*n) / Product_{k=1..n} (1-x^k).
(2) A(x) = Sum_{n>=0} x^(n*(3n+1)/2)*A(x)^(2*n)*(1 + x^(2n+1)*A(x)^2)*Product_{k=1..n} (1 + x^k*A(x)^2)/(1-x^k) due to Sylvester's identity.
a(n) ~ c * d^n / n^(3/2), where d = 5.5051727555189932106045782067309509... and c = 0.4987046473347092789085107139372... - Vaclav Kotesovec, Sep 28 2023
Radius of convergence r = 0.181647342310464199522927295317... = 1/d and A(r) = 1.82512871645978495662055342941... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 2*r^n/(1 + r^n*A(r)^2) ) and A(r) = Product_{n>=1} (1 + r^n*A(r)^2). - Paul D. Hanna, Mar 03 2024
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 179*x^5 + 778*x^6 + ...
where
(0) A(x) = (1+x*A(x)^2) * (1+x^2*A(x)^2) * (1+x^3*A(x)^2) * (1+x^4*A(x)^2) * ...
(1) A(x) = 1 + x*A(x)^2/(1-x) + x^3*A(x)^4/((1-x)*(1-x^2)) + x^6*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + x^10*A(x)^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + ...
(2) A(x) = (1+x*A(x)^2) + x^2*A(x)^2*(1 + x^3*A(x)^2)*(1+x*A(x)^2)/(1-x) + x^7*A(x)^4*(1 + x^5*A(x)^2)*(1+x*A(x)^2)*(1+x^2*A(x)^2)/((1-x)*(1-x^2)) + x^15*A(x)^6*(1 + x^7*A(x)^2)*(1+x*A(x)^2)*(1+x^2*A(x)^2)*(1+x^3*A(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, {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*(1 + s^2) * Derivative[0, 1][QPochhammer][-s^2, r] / (8*Pi*(s^2 + (1 + s^2)^2 * QPolyGamma[1, Log[-s^2]/Log[r], r]/ Log[r]^2))]} /. FindRoot[{QPochhammer[-s^2, r] == s*(1 + s^2), 1/2 + s^2/(1 + s^2) + (Log[1 - r] + QPolyGamma[0, Log[-s^2]/Log[r], r])/Log[r] == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 28 2023 *)
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PROG
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(PARI) {a(n) = my(A=1+x); for(i=1, n, A=prod(m=1, n, (1+A^2*x^m+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n) = my(A=1+x); for(i=1, n, A=sum(m=0, n, x^(m*(m+1)/2)*A^(2*m)/prod(k=1, m, 1-x^k +x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n) = my(A=1+x); for(i=1, n, A=sum(m=0, n, x^(m*(3*m+1)/2)*A^(2*m)*(1 + x^(2*m+1)*A^2)*prod(k=1, m, (1+A^2*x^k)/(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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