A192626 G.f. satisfies: A(x) = Product_{n>=0} (1 + x*(x+x^2)^n)^2/(1 - x*(x+x^2)^n)^2.

1, 4, 12, 36, 100, 264, 676, 1684, 4096, 9764, 22888, 52872, 120540, 271600, 605556, 1337320, 2927720, 6358432, 13707916, 29351536, 62450468, 132090356, 277845120, 581405140, 1210688864, 2509483020, 5178969644, 10644112012, 21790816340, 44444609044

Offset: 0

Author

Paul D. Hanna, Jul 16 2011

Keywords

• nonn

Comments

Related q-series identity due to Heine:

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^2, x=x, y=z=1.

Examples

			G.f.: A(x) = 1 + 4*x + 12*x^2 + 36*x^3 + 100*x^4 + 264*x^5 + 676*x^6 +...
where the g.f. equals the product:
A(x) = (1+x)^2/(1-x)^2 * (1+x^2*(1+x))^2/(1-x^2*(1+x))^2 * (1+x^3*(1+x)^2)^2/(1-x^3*(1+x)^2)^2 * (1+x^4*(1+x)^3)^2/(1-x^4*(1+x)^3)^2 *...
which is also equal to the sum:
A(x) = 1 + 4*x/((1-x)*(1-x*(1+x))) + 4*x^2*(1+x*(1+x))^2/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)) + 4*x^3*(1+x*(1+x))^2*(1 + x^2*(1+x)^2)^2/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)*(1-x^3*(1+x)^2)*(1-x^3*(1+x)^3)) +...
		

Crossrefs

Cf. A192627, A192622.

Programs

PARI

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

PARI

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

Formula

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

Self-convolution of A192627.