A192622 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)*A(x)^n)^2/(1 - x^(n+1)*A(x)^n)^2.

1, 4, 12, 48, 220, 1080, 5600, 30112, 166300, 937620, 5374200, 31221488, 183430656, 1087975256, 6505878592, 39179738400, 237412139260, 1446488046824, 8855937880108, 54455375407504, 336159421649528, 2082508824181856, 12942736191473792

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*A(x), x=x, y=z=1.

Links

Table of n, a(n) for n=0..22.

Formula

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

Self-convolution of A192623.

Example

G.f.: A(x) = 1 + 4*x + 12*x^2 + 48*x^3 + 220*x^4 + 1080*x^5 +...
The g.f. A = A(x) satisfies the following relations:
A = (1+x)^2/(1-x)^2 * (1+x^2*A)^2/(1-x^2*A)^2 * (1+x^3*A^2)^2/(1-x^3*A^2)^2 *...
A = 1 + 4*x/((1-x)*(1-x*A)) + 4*x^2*(1+x*A)^2/((1-x)*(1-x*A)*(1-x^2*A)*(1-x^2*A^2)) + 4*x^3*(1+x*A)^2*(1+x^2*A^2)^2/((1-x)*(1-x*A)*(1-x^2*A)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x^3*A^3)) +...

Prog

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

Crossrefs

Cf. A192623, A192620, A192624.

Sequence in context: A110594 A151483 A355800 * A324801 A111930 A335595

Adjacent sequences: A192619 A192620 A192621 * A192623 A192624 A192625

Keyword

nonn

Author

Paul D. Hanna, Jul 06 2011

Status

approved