A192624 G.f. satisfies: A(x) = Product_{n>=1} (1+x^n)*(1 + x^n*A(x))/((1-x^n)*(1 - x^n*A(x))).

1, 4, 20, 112, 676, 4328, 28912, 199392, 1409364, 10157828, 74375640, 551715264, 4137527408, 31318286632, 238958947328, 1835960454272, 14192132860868, 110298595778872, 861338925309604, 6755283201399776, 53185599585579640

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=A(x), z=1.

Links

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

Formula

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

Example

G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4328*x^5 +...
The g.f. A = A(x) satisfies:
A = (1+x)*(1+x*A)/((1-x)*(1-x*A)) * (1+x^2)*(1+x^2*A)/((1-x^2)*(1-x^2*A)) * (1+x^3)*(1+x^3*A)/((1-x^3)*(1-x^3*A)) *...
A = {1 + 2*x*(A+1)/(1-x)^2 + 2*x^2*(A+1)*(A+x)*(1+x)/((1-x)*(1-x^2))^2 + 2*x^3*(A+1)*(A+x)*(A+x^2)*(1+x)*(1+x^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)*(1+x^k)/((1-x^k+x*O(x^n))*(1-x^k*A)))); 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)*(1+x^k)/(1-x^(k+1)+x*O(x^n))^2))); polcoeff(A, n)}

Crossrefs

Cf. A192625, A192620, A192622.

Sequence in context: A003645 A081085 A212326 * A209200 A294119 A245375

Adjacent sequences: A192621 A192622 A192623 * A192625 A192626 A192627

Keyword

nonn

Author

Paul D. Hanna, Jul 06 2011

Status

approved