A192627 G.f. satisfies: A(x) = Product_{n>=0} (1 + x*(x+x^2)^n)/(1 - x*(x+x^2)^n).

1, 2, 4, 10, 22, 48, 104, 222, 466, 966, 1988, 4060, 8222, 16528, 33024, 65620, 129698, 255096, 499508, 974032, 1891866, 3661034, 7060324, 13572010, 26009822, 49701946, 94714606, 180022550, 341316642, 645594510, 1218377230, 2294387492, 4311757732

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=1, z=0.

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 22*x^4 + 48*x^5 + 104*x^6 +...
where the g.f. equals the product:
A(x) = (1+x)/(1-x) * (1+x^2*(1+x))/(1-x^2*(1+x)) * (1+x^3*(1+x)^2)/(1-x^3*(1+x)^2) * (1+x^4*(1+x)^3)/(1-x^4*(1+x)^3) *...
which is also equal to the sum:
A(x) = 1 + 2*x/((1-x)*(1-x*(1+x))) + 2*x^3*(1+x)*(1+x*(1+x))/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)) + 2*x^6*(1+x)*(1+x*(1+x))*(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. A192626, A192623.

Programs

PARI

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

PARI

{a(n)=local(A=1+x);A=1+sum(m=1,n,x^m*(x+x^2)^(m*(m-1)/2)*prod(k=0,m-1,(1+(x+x^2)^k)/((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*(x+x^2)^(n*(n-1)/2) * Product_{k=0..n-1} (1+(x+x^2)^k)/((1-x*(x+x^2)^k)*(1-(x+x^2)^(k+1)) due to the Heine identity.

Self-convolution equals A192626.