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A192502 G.f. satisfies: A(x) = 1 + x*f(x, A(x)) where f(,) is Ramanujan's two-variable theta function. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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)*...

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

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Last modified December 4 09:25 EST 2020. Contains 338921 sequences. (Running on oeis4.)