A246876 G.f.: 1 / AGM(1-12*x, sqrt((1-4*x)*(1-36*x))).

1, 16, 324, 7744, 206116, 5875776, 175191696, 5386385664, 169300977444, 5410164352576, 175128910042384, 5727842622630144, 188931648862083856, 6276176070222305536, 209747841324097564224, 7046053064278540084224, 237764385841359952067364, 8054915184317632144620096

Offset: 0

Author

Paul D. Hanna, Sep 06 2014

Keywords

• nonn

Comments

In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by

1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*[Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k)]^2,

and consists of integer coefficients when 4|(q-p).

Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 16*x + 324*x^2 + 7744*x^3 + 206116*x^4 + 5875776*x^5 +...
where the square-root of the terms yields A081671:
[1, 4, 18, 88, 454, 2424, 13236, 73392, 411462, 2325976, ...]
the g.f. of which is 1/sqrt((1-2*x)*(1-6*x)).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..644

Crossrefs

Cf. A081671, A168597, A246467.

Programs

PARI

{a(n)=polcoeff( 1 / agm(1-12*x, sqrt((1-4*x)*(1-36*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n)=sum(k=0,n,2^(n-k)*binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = A081671(n)^2 = [Sum_{k=0..n} 2^(n-k) * C(n,k) * C(2*k,k)]^2.

G.f.: 1 / AGM((1-2*x)*(1+6*x), (1+2*x)*(1-6*x)) = Sum_{n>=0} a(n)*x^(2*n).

a(n) ~ 2^(2*n - 1) * 3^(2*n + 1) / (Pi*n). - Vaclav Kotesovec, Dec 10 2018