A246906 G.f.: 1 / AGM(1-21*x, sqrt((1-9*x)*(1-49*x))).

1, 25, 729, 24025, 866761, 33350625, 1342856025, 55849505625, 2378365418025, 103099146750625, 4531090723144129, 201324497403240225, 9025111586043157801, 407581475160408424225, 18521763259935613598649, 846187436813348419025625, 38838031986984135802130025

Offset: 0

Author

Paul D. Hanna, Sep 07 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 + 25*x + 729*x^2 + 24025*x^3 + 866761*x^4 +...
where the square-root of the terms yields A098409:
[1, 5, 27, 155, 931, 5775, 36645, 236325, 1542195, ...],
the g.f. of which is 1/sqrt((1-3*x)*(1-7*x)).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..593

Crossrefs

Cf. A098409, A246467, A248167.

Programs

Mathematica

a[n_] := Sum[3^(n - k) * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]^2; Array[a, 17, 0] (* Amiram Eldar, Dec 11 2018 *)

PARI

{a(n,p=3,q=7)=polcoeff( 1 / agm(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x) +x*O(x^n))), n) }
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n,p=3,q=7)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n,p=3,q=7)=sum(k=0,n,p^(n-k)*((q-p)/4)^k*binomial(n,k)*binomial(2*k,k))^2 }
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = A098409(n)^2 = [Sum_{k=0..n} 3^(n-k)*C(n,k)*C(2*k,k)]^2.

G.f.: 1 / AGM((1-3*x)*(1+7*x), (1+3*x)*(1-7*x)) = Sum_{n>=0} a(n)*x^(2*n).

From Vaclav Kotesovec, Sep 27 2019: (Start)

Recurrence: n^2*(2*n - 3)*a(n) = (2*n - 1)*(79*n^2 - 158*n + 54)*a(n-1) - 21*(2*n - 3)*(79*n^2 - 158*n + 54)*a(n-2) + 9261*(n-2)^2*(2*n - 1)*a(n-3).

a(n) ~ 7^(2*n+1) / (4*Pi*n). (End)