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A192625 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/((1-x^n)*(1 - x^n*A(x)^2)). 2
1, 4, 28, 240, 2348, 24952, 280192, 3271232, 39310668, 483032980, 6041149272, 76648727632, 984161689728, 12764078032568, 166969699620640, 2200415358484800, 29186416580736300, 389340777798701672, 5220028320540100220, 70303231772070200912 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Related q-series (Heine) identity:

1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)),

here q=x, x=x, y=z=A(x).

LINKS

Table of n, a(n) for n=0..19.

FORMULA

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (A(x) + x^k)^2/(1-x^(k+1))^2 due to the Heine identity.

EXAMPLE

G.f.: A(x) = 1 + 4*x + 28*x^2 + 240*x^3 + 2348*x^4 + 24952*x^5 +...

The g.f. A = A(x) satisfies:

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

A = {1 + x*(A+1)^2/(1-x)^2 + x^2*(A+1)^2*(A+x)^2/((1-x)*(1-x^2))^2 + x^3*(A+1)^2*(A+x)^2*(A+x^2)^2/((1-x)*(1-x^2)*(1-x^3))^2 +...

PROG

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

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*prod(k=0, m-1, (A+x^k)^2/(1-x^(k+1)+x*O(x^n))^2))); polcoeff(A, n)}

CROSSREFS

Cf. A192624, A192620, A192622.

Sequence in context: A354693 A112113 A188266 * A199561 A103211 A229644

Adjacent sequences:  A192622 A192623 A192624 * A192626 A192627 A192628

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 06 2011

STATUS

approved

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Last modified August 17 19:38 EDT 2022. Contains 356189 sequences. (Running on oeis4.)