A243944 a(n) = A084768(n)^2.

1, 49, 5329, 717409, 106523041, 16735820689, 2727812288881, 456250924320961, 77788137919752001, 13459803510972477169, 2356471368269511061009, 416518496068852312607521, 74207592486779379593752801, 13309569813247406938272432721, 2400816685486139045360488325809

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*(t+1))^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 A084768 is 1/sqrt(1 - 14*x + x^2).

Limit a(n+1)/a(n) = (7 + 4*sqrt(3))^2 = 97 + 56*sqrt(3).

Examples

			G.f.: A(x) = 1 + 49*x + 5329*x^2 + 717409*x^3 + 106523041*x^4 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..438

Crossrefs

Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), A243949 (m=1), A243943 (m=2), this sequence (m=3), A243007 (m=4).

Cf. A084768.

Programs

Magma

[Evaluate(LegendrePolynomial(n),7)^2 : n in [0..40]]; // G. C. Greubel, May 17 2023

Mathematica

Table[Sum[12^k * Binomial[2*k, k]^2 * Binomial[n+k, n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 28 2019 *)
CoefficientList[Series[2*EllipticK[1 - (1-x)^2/(1 - 194*x + x^2)] / (Pi*Sqrt[1 - 194*x + x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 28 2019 *)
LegendreP[Range[0, 40], 7]^2 (* G. C. Greubel, May 17 2023 *)

PARI

{a(n) = sum(k=0, n, 12^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, 3^k * binomial(2*k, k) * binomial(n+k, n-k) )^2}
for(n=0, 20, print1(a(n), ", "))

PARI

/* Using AGM: */
{a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 14^2*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 30 2014

SageMath

[gen_legendre_P(n,0,7)^2 for n in range(41)] # G. C. Greubel, May 17 2023

Formula

G.f.: 1 / AGM(1-x, sqrt(1-194*x+x^2)). - Paul D. Hanna, Aug 30 2014

a(n) = Sum_{k=0..n} 12^k * C(2*k, k)^2 * C(n+k, n-k).

a(n)^(1/2) = Sum_{k=0..n} 3^k * C(2*k, k) * C(n+k, n-k).

a(n) ~ (1+sqrt(3))^(8*n+4) / (sqrt(3) * Pi * n * 2^(4*n+5)). - Vaclav Kotesovec, Sep 28 2019

a(n) = (LegendreP(n, 7))^2. - G. C. Greubel, May 17 2023