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A200222 G.f. satisfies: A(x) = 1 + Sum_{n>=1} (x*A(x))^(n^2) * (A(x)^n + 1/A(x)^n). 1
1, 2, 4, 12, 42, 164, 688, 3024, 13680, 63110, 295520, 1401012, 6713280, 32470468, 158343504, 777725264, 3843992546, 19104857608, 95419519076, 478668009828, 2410698765472, 12184259877320, 61782045169312, 314202878599696, 1602270787137472, 8191160756085318, 41971595130249968, 215522156779513584 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..430 (terms 0..200 from Paul D. Hanna)

FORMULA

By the Jacobi triple product identity, g.f. A(x) satisfies:

(1) A(x) = Product_{n>=1} (1 + x^(2*n-1)*A(x)^(2*n-2)) * (1 + x^(2*n-1)*A(x)^(2*n)) * (1 - x^(2*n)*A(x)^(2*n)).

Let G(x) be the g.f. of A190791, then A(x) satisfies:

(2) A(x) = (1/x)*Series_Reversion(x/G(x)),

(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),

(4) a(n) = [x^n] G(x)^(n+1)/(n+1),

where G(x) = 1 + Sum_{n>=1} x^(n^2) * (G(x)^n + 1/G(x)^n).

a(n) ~ c * d^n / n^(3/2), where d = 5.42800145666083947972618... and c = 0.45497910593346577587... - Vaclav Kotesovec, Sep 04 2017

EXAMPLE

G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 42*x^4 + 164*x^5 + 688*x^6 +...

The g.f. A = A(x) may be expressed by the series:

A(x) = 1 + x*A*(A + 1/A) + x^4*A^4*(A^2 + 1/A^2) + x^9*A^9*(A^3 + 1/A^3) + x^16*A^16*(A^4 + 1/A^4) + x^25*A^25*(A^5 + 1/A^5) +...

and by the Jacobi triple product:

A(x) = (1+x)*(1+x*A^2)*(1-x^2*A^2) * (1+x^3*A^2)*(1+x^3*A^4)*(1-x^4*A^4) * (1+x^5*A^4)*(1+x^5*A^6)*(1-x^6*A^6) * (1+x^7*A^6)*(1+x^7*A^8)*(1-x^8*A^8) *...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), (A^m +1/(A+x*O(x^n))^m)*(x*A)^(m^2))); polcoeff(A, n)}

(PARI) /* By Jacobi's Triple Product Identity: */

{a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n\2+1, (1+x^(2*m-1)*A^(2*m-2)+x*O(x^n))*(1+x^(2*m-1)*A^(2*m))*(1-x^(2*m)*A^(2*m)) )); polcoeff(A, n)}

CROSSREFS

Cf. A190791, A199576, A196354, A197707.

Sequence in context: A126947 A188494 A202668 * A063179 A096802 A139669

Adjacent sequences:  A200219 A200220 A200221 * A200223 A200224 A200225

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 14 2011

STATUS

approved

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Last modified February 17 18:14 EST 2020. Contains 332005 sequences. (Running on oeis4.)