OFFSET
0,2
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).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..438
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
EXAMPLE
G.f.: A(x) = 1 + 49*x + 5329*x^2 + 717409*x^3 + 106523041*x^4 +...
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 *)
PROG
(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
(Magma) [Evaluate(LegendrePolynomial(n), 7)^2 : n in [0..40]]; // G. C. Greubel, May 17 2023
(SageMath) [gen_legendre_P(n, 0, 7)^2 for n in range(41)] # G. C. Greubel, May 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 18 2014
STATUS
approved