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A243946 G.f.: sqrt( (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)) ). 6
1, 7, 91, 1345, 20995, 337877, 5544709, 92234527, 1549694195, 26237641045, 446925926881, 7650344197987, 131489964887341, 2267722252458475, 39224201631222475, 680160975405238145, 11820134678459908115, 205812328555924135045, 3589742656727603141425, 62707329988264214752675 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Square each term to form a bisection of A243945.

Limit a(n+1)/a(n) = 9 + 4*sqrt(5).

LINKS

Table of n, a(n) for n=0..19.

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

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

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 *)

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), ", "))

CROSSREFS

Cf. A243945, A243947, A084769, A245926, A008316.

Sequence in context: A165230 A156712 A004368 * A130978 A191097 A234570

Adjacent sequences:  A243943 A243944 A243945 * A243947 A243948 A243949

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Aug 17 2014

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)