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A192621 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^2)/(1 - x^n*A(x)^2). 3
1, 2, 12, 88, 726, 6456, 60392, 585792, 5838764, 59440250, 615431464, 6460681656, 68607630680, 735682014648, 7954732578032, 86635206695808, 949518438959574, 10464751843723840, 115904823140622164, 1289419736206548408, 14401729960605163272 (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*A(x)^2, y=1, z=0.

LINKS

Table of n, a(n) for n=0..20.

FORMULA

G.f. A(x) satisfies:

(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))/(1-x^k), due to the q-binomial theorem.

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

(3) A(x)^2 = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x)^2)*(1-x^k), due to the Heine identity.

Self-convolution yields A192620.

EXAMPLE

G.f.: A(x) = 1 + 2*x + 12*x^2 + 88*x^3 + 726*x^4 + 6456*x^5 +...

The g.f. A = A(x) satisfies the following relations:

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

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

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

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

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*A^2+x*O(x^n)))); polcoeff(A, n)}

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

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

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

CROSSREFS

Cf. A192620 (g.f. A(x)^2), A192623, A190862.

Sequence in context: A305868 A319324 A059435 * A143923 A079858 A224152

Adjacent sequences:  A192618 A192619 A192620 * A192622 A192623 A192624

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 06 2011

STATUS

approved

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Last modified September 19 06:56 EDT 2021. Contains 347551 sequences. (Running on oeis4.)