A192502 G.f. satisfies: A(x) = 1 + x*f(x, A(x)) where f(,) is Ramanujan's two-variable theta function.

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

Offset

0,2

Comments

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).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Michael Somos, Introduction to Ramanujan theta functions.

Formula

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

Example

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)*...

Mathematica

(* 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 *)

Prog

(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)}

Crossrefs

Sequence in context: A188495 A038077 A006396 * A243567 A323949 A066278

Adjacent sequences: A192499 A192500 A192501 * A192503 A192504 A192505

Keyword

nonn

Author

Paul D. Hanna, Jul 03 2011

Status

approved