A243007 a(n) = A084769(n)^2.

1, 81, 14641, 3272481, 806616801, 210358905201, 56912554609681, 15800522430616641, 4471485120646226881, 1284238494711502355601, 373195323236525968732401, 109489964937514282794301281, 32378265673661271315300820641, 9639042117142706280223219663281

Offset: 0

Author

Paul D. Hanna, Aug 18 2014

Keywords

• nonn

Comments

In general, we have the binomial identity:

if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k),

then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k),

where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2),

and the g.f. of b(n)^2 is 1 / AGM(1-x, sqrt((1+x)^2 - (4*t+2)^2*x)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Note that the g.f. of A084769 is 1/sqrt(1 - 18*x + x^2).

Limit_{n -> oo} a(n+1)/a(n) = (9 + 4*sqrt(5))^2 = 161 + 72*sqrt(5).

Examples

			G.f.: A(x) = 1 + 81*x + 14641*x^2 + 3272481*x^3 + 806616801*x^4 +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), A243949 (m=1), A243943 (m=2), A243944 (m=3), this sequence (m=4).

Cf. A084769.

Programs

Magma

[Evaluate(LegendrePolynomial(n),9)^2 : n in [0..30]]; // G. C. Greubel, May 17 2023

Mathematica

Table[SeriesCoefficient[1/Sqrt[1 -18x +x^2], {x,0,n}], {n,0,20}]^2 (* Vincenzo Librandi, Feb 14 2018 *)
LegendreP[Range[0,30], 9]^2 (* G. C. Greubel, May 17 2023 *)

PARI

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

PARI

{a(n) = sum(k=0, n, 4^k * binomial(2*k, k) * binomial(n+k, n-k) )^2}
for(n=0, 20, print1(a(n), ", "))
{a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 18^2*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 30 2014

SageMath

[gen_legendre_P(n,0,9)^2 for n in range(41)] # G. C. Greubel, May 17 2023

Formula

G.f.: 1 / AGM(1-x, sqrt(1- 322*x + x^2)). - Paul D. Hanna, Aug 30 2014

a(n) = Sum_{k=0..n} 20^k * C(2*k, k)^2 * C(n+k, n-k).

a(n)^(1/2) = Sum_{k=0..n} 4^k * C(2*k, k) * C(n+k, n-k).

a(n) ~ (2 + sqrt(5))^(4*n+2) / (8*sqrt(5)*Pi*n). - Vaclav Kotesovec, Sep 28 2019