A246467 G.f.: 1 / AGM(1-5*x, sqrt((1-x)*(1-25*x))).

1, 9, 121, 2025, 38025, 762129, 15912121, 341621289, 7484845225, 166549691025, 3751508008161, 85341068948529, 1957289174870121, 45199191579030225, 1049893021288265625, 24510327614556266025, 574726636455361317225, 13528549573868347823025, 319541915502909478890625

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 + 9*x + 121*x^2 + 2025*x^3 + 38025*x^4 + 762129*x^5 +...
where the square-root of the terms yields A026375:
[1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, ...]
the g.f. of which is 1/sqrt((1-x)*(1-5*x)).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..717

Crossrefs

Cf. A026375, A168597, A246876, A246906, A248167.

Programs

Mathematica

CoefficientList[Series[1/ArithmeticGeometricMean[1-5x,Sqrt[(1-x)(1-25x)]],{x,0,20}],x] (* Harvey P. Dale, Nov 01 2023 *)

PARI

{a(n)=polcoeff( 1 / agm(1-5*x, sqrt((1-x)*(1-25*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

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

Formula

a(n) = A026375(n)^2 = [Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)]^2.

G.f.: 1 / AGM((1-x)*(1+5*x), (1+x)*(1-5*x)) = Sum_{n>=0} a(n)*x^(2*n).

a(n) ~ 5^(2*n+1) / (4*Pi*n). - Vaclav Kotesovec, Dec 10 2018