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A192502
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G.f. satisfies: A(x) = 1 + x*f(x, A(x)) where f(,) is Ramanujan's two-variable theta function.
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2
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1, 2, 4, 10, 36, 136, 548, 2316, 10050, 44426, 199666, 910090, 4196984, 19545844, 91791112, 434181656, 2066656564, 9891669820, 47578282002, 229858639366, 1114895656402, 5427058308018, 26503888167186, 129821343271168, 637626106479490
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OFFSET
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0,2
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COMMENTS
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Ramanujan's two-variable theta function is defined by:
f(a,b) = Sum_{n=-infinity..+infinity} a^(n*(n+1)/2) * b^(n*(n-1)/2).
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x) = 1+x + x*Sum_{n>=1} (x*A(x))^(n*(n-1)/2) * (x^n + A(x)^n).
(2) A(x) = 1 + x*Product_{n>=0} (1+x*q^n)*(1+A(x)*q^n)*(1-q^(n+1)) where q=x*A(x), due to Jacobi's triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 5.2286591857647664516287778... and c = 0.4431871616898705063582... - Vaclav Kotesovec, Sep 04 2017
Formula (2) can be rewritten as the functional equation y = 1 + x*QPochhammer(-x, x*y) * QPochhammer(-y, x*y) * QPochhammer(x*y). - Vaclav Kotesovec, Jan 19 2024
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 36*x^4 + 136*x^5 + 548*x^6 +...
The g.f. A = A(x) satisfies:
(1) A = 1+x + x*[(x+A) + x*A*(x^2+A^2) + x^3*A^3*(x^3+A^3) + x^6*A^6*(x^4+A^4) + x^10*A^10*(x^5+A^5) +...].
(2) A = 1 + x*(1+x)*(1+A)*(1-x*A)* (1+x^2*A)*(1+x*A^2)*(1-x^2*A^2)* (1+x^3*A^2)*(1+x^2*A^3)*(1-x^3*A^3)* (1+x^4*A^3)*(1+x^3*A^4)*(1-x^4*A^4)*...
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MATHEMATICA
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(* Calculation of constant d: *) 1/r /. FindRoot[{s == 1 + r*QPochhammer[-r, r*s] * QPochhammer[-s, r*s] * QPochhammer[r*s], r*(-1 + s) * Derivative[0, 1][QPochhammer][-r, r*s] / QPochhammer[-r, r*s] + r^2*QPochhammer[-r, r*s] * QPochhammer[r*s] * Derivative[0, 1][QPochhammer][-s, r*s] + (-1 + s)*(-((2*Log[1 - r*s] + QPolyGamma[0, 1, r*s] + QPolyGamma[0, Log[-s]/Log[r*s], r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s] / QPochhammer[r*s]) == 1}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x+x*sum(m=1, sqrtint(2*n)+1, (x*A+x*O(x^n))^(m*(m-1)/2)*(x^m+A^m))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, q=x*(A+O(x^n)); A=1+x*prod(m=0, n, (1+x*q^m)*(1+A*q^m)*(1-q^(m+1))) ); 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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