OFFSET
0,2
COMMENTS
Square each term to form a bisection of A243945.
Limit_{n->oo} a(n+1)/a(n) = 9 + 4*sqrt(5).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500
FORMULA
a(n)^2 = Sum_{k=0..2*n} C(2*k, k)^2 * C(2*n+k, 2*n-k).
a(n) ~ sqrt(2+sqrt(5)) * (9+4*sqrt(5))^n / (2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Aug 18 2014. Equivalently, a(n) ~ phi^(6*n + 3/2) / (2*sqrt(2*Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
From Peter Bala, Mar 14 2018: (Start)
a(n) = P(2*n,sqrt(5)), where P(n,x) denotes the n-th Legendre polynomial. See A008316.
a(n) = (1/C(2*n,n))*Sum_{k = 0..n} C(n,k)*C(n+k,k)* C(2*n+2*k,n+k). In general, P(2*n,sqrt(1 + 4*x)) = (1/C(2*n,n))*Sum_{k=0..n} C(n,k)*C(n+k,k)*C(2*n+2*k,n+k)*x^k.
a(n) = Sum_{k = 0..2*n} C(2*n,k)^2 * phi^(2*n-2*k), where phi = (sqrt(5) + 1)/2.
a(n) = Sum_{k = 0..2*n} C(2*n,k)*C(2*n+k,k)*Phi^k, where Phi = (sqrt(5) - 1)/2. (End)
a(n) = hypergeom([-n, n + 1/2], [1], -4). - Peter Luschny, Mar 16 2018
D-finite with recurrence: n*(2*n-1)*(4*n-5)*a(n) -(4*n-3)*(36*n^2-54*n+11)*a(n-1) +(n-1)*(4*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020
a(n) = Sum_{k=0..n} 5^(n-k) * binomial(2*k,k) * binomial(2*n,2*k). - Seiichi Manyama, Aug 25 2020
EXAMPLE
G.f.: A(x) = 1 + 7*x + 91*x^2 + 1345*x^3 + 20995*x^4 + 337877*x^5 + ...,
where A(x)^2 = (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)).
MAPLE
seq(add(binomial(n, k)*binomial(n+k, k)*binomial(2*n+2*k, n+k), k = 0..n)/binomial(2*n, n), n = 0..20); # Peter Bala, Mar 14 2018
MATHEMATICA
a[n_] := Hypergeometric2F1[-n, n + 1/2, 1, -4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 16 2018 *)
CoefficientList[Series[Sqrt[(1+x+Sqrt[1-18x+x^2])/(2(1-18x+x^2))], {x, 0, 20}], x] (* Harvey P. Dale, Dec 26 2019 *)
a[n_] := Sum[(5^k Gamma[2 n + 1])/(Gamma[2 k + 1]*Gamma[n - k + 1]^2), {k, 0, n}];
Flatten[Table[a[n], {n, 0, 19}]] (* Detlef Meya, May 22 2024 *)
PROG
(PARI) /* From definition: */
{a(n)=polcoeff( sqrt( (1+x + sqrt(1-18*x+x^2 +x*O(x^n))) / (2*(1-18*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From a(n) = sqrt( A243945(2*n) ): */
{a(n)=sqrtint( sum(k=0, 2*n, binomial(2*k, k)^2*binomial(2*n+k, 2*n-k)) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, 5^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))} \\ Seiichi Manyama, Aug 25 2020
(Python)
from math import comb
def A243946(n): return sum(5**(n-k)*comb(m:=k<<1, k)*comb(n<<1, m) for k in range(n+1)) # Chai Wah Wu, Mar 23 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Aug 17 2014
STATUS
approved