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A190791 G.f. satisfies: A(x) = 1 + Sum_{n>=1} (A(x)^n + A(x)^-n) * x^(n^2). 7
1, 2, 0, 4, -6, 32, -88, 376, -1376, 5574, -22232, 91548, -378736, 1589304, -6719040, 28647592, -122933470, 530755764, -2303432600, 10043949684, -43979901840, 193309672224, -852599615912, 3772221225128, -16737583019616, 74461240879386 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

FORMULA

G.f. satisfies: A(x) = Product_{n>=1} (1 - x^(2n))*(1 + x^(2n-1)*A(x))*(1 + x^(2n-1)/A(x)), due to the Jacobi triple product identity.

a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 4.73097028144959... and c = 0.1236197969613... . - Vaclav Kotesovec, Mar 02 2016

EXAMPLE

G.f.: A(x) = 1 + 2*x + 4*x^3 - 6*x^4 + 32*x^5 - 88*x^6 + 376*x^7 +...

The g.f. A(x) satisfies the series:

* A(x) = 1 + (A(x) + A(x)^-1)*x + (A(x)^2 + A(x)^-2)*x^4 + (A(x)^3 + A(x)^-3)*x^9 + (A(x)^4 + A(x)^-4)*x^16 +...

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

which is a result due to the Jacobi triple product identity.

PROG

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

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

CROSSREFS

Sequence in context: A287846 A085623 A317965 * A002885 A011121 A255643

Adjacent sequences:  A190788 A190789 A190790 * A190792 A190793 A190794

KEYWORD

sign

AUTHOR

Paul D. Hanna, May 20 2011

STATUS

approved

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Last modified October 23 05:56 EDT 2019. Contains 328335 sequences. (Running on oeis4.)