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^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).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
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
EXAMPLE
G.f.: A(x) = 1 + 81*x + 14641*x^2 + 3272481*x^3 + 806616801*x^4 +...
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 *)
PROG
(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
(Magma) [Evaluate(LegendrePolynomial(n), 9)^2 : n in [0..30]]; // G. C. Greubel, May 17 2023
(SageMath) [gen_legendre_P(n, 0, 9)^2 for n in range(41)] # G. C. Greubel, May 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 18 2014
STATUS
approved