A245927 G.f.: sqrt( (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)) ).

1, 9, 99, 1175, 14499, 183195, 2351805, 30539241, 400000275, 5274560891, 69929215641, 931226954949, 12446852889901, 166888293332805, 2243683808486451, 30235162687458327, 408274269493595283, 5523024440001832875, 74834275541765522505, 1015429462194625633125

Offset: 0

Author

Paul D. Hanna, Aug 15 2014

Keywords

• nonn
• easy

Comments

Multiply the square of each term by -3 to form a bisection of A245925.

Limit a(n+1)/a(n) = 7 + 4*sqrt(3).

Examples

			G.f.: A(x) = 1 + 9*x + 99*x^2 + 1175*x^3 + 14499*x^4 + 183195*x^5 +... where
A(x)^2 = (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)). Explicitly,
A(x)^2 = 1 + 18*x + 279*x^2 + 4132*x^3 + 59949*x^4 + 860022*x^5 + 12252547*x^6 + 173756232*x^7 + 2456093529*x^8 +...+ A245924(n)*x^n +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..850

Crossrefs

Cf. A245925, A245926, A245924, A243947.

Programs

Maple

seq(add( binomial(2*n+1,2*k)*binomial(2*k,k)*3^(n-k), k = 0..n),n = 0..20); # Peter Bala, Mar 17 2018

Mathematica

CoefficientList[Series[Sqrt[(1-x-Sqrt[1-14x+x^2])/(6x(1-14x+x^2))],{x,0,20}],x] (* Harvey P. Dale, Oct 23 2015 *)
a[n_] := (-1)^n Hypergeometric2F1[-n, n + 3/2, 1, 4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 17 2018 *)

PARI

/* From definition: */
{a(n)=polcoeff( sqrt( (1-x - sqrt(1-14*x+x^2 +x^2*O(x^n))) / (6*x*(1-14*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

/* From formula for a(n)^2: */
{a(n)=sqrtint((-1/3)*sum(k=0, 2*n+1, sum(j=0, 4*n-2*k+2, (-1)^(j+k)*binomial(4*n-k+2,j+k)^2*binomial(j+k, k)^2)))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n)^2 = (-1/3)*Sum_{k=0..2*n+1} Sum_{j=0..4*n-2*k+2} (-1)^(j+k) * C(4*n-k+2,j+k)^2 * C(j+k,k)^2.

a(n) ~ (3-sqrt(3)) * (7+4*sqrt(3))^(n+1) / (12*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 16 2014

From Peter Bala, Mar 17 2018: (Start)

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

a(n) = 3^n*hypergeom([-n, -n-1/2], [1], 4/3).

n*(4*n-3)*(2*n+1)*a(n) = (4*n-1)*(28*n^2-14*n-5)*a(n-1) - (n-1)*(2*n-1)*(4*n+1)*a(n-2). (End)

a(n) = (-1)^n*hypergeom([-n, n + 3/2], [1], 4). Peter Luschny, Mar 17 2018

a(n) = (-1)^n/sqrt(-3) * P(2*n+1, sqrt(-3)), where P(n, x) denotes the n-th Legendre polynomial. - Peter Bala, Mar 17 2024