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Expansion of sqrt( (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)) ).
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%I #42 May 23 2024 15:14:29

%S 1,7,91,1345,20995,337877,5544709,92234527,1549694195,26237641045,

%T 446925926881,7650344197987,131489964887341,2267722252458475,

%U 39224201631222475,680160975405238145,11820134678459908115,205812328555924135045,3589742656727603141425,62707329988264214752675

%N Expansion of sqrt( (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)) ).

%C Square each term to form a bisection of A243945.

%C Limit_{n->oo} a(n+1)/a(n) = 9 + 4*sqrt(5).

%H Seiichi Manyama, <a href="/A243946/b243946.txt">Table of n, a(n) for n = 0..500</a>

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

%F 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

%F From _Peter Bala_, Mar 14 2018: (Start)

%F a(n) = P(2*n,sqrt(5)), where P(n,x) denotes the n-th Legendre polynomial. See A008316.

%F 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.

%F a(n) = Sum_{k = 0..2*n} C(2*n,k)^2 * phi^(2*n-2*k), where phi = (sqrt(5) + 1)/2.

%F 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)

%F a(n) = hypergeom([-n, n + 1/2], [1], -4). - _Peter Luschny_, Mar 16 2018

%F 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

%F a(n) = Sum_{k=0..n} 5^(n-k) * binomial(2*k,k) * binomial(2*n,2*k). - _Seiichi Manyama_, Aug 25 2020

%e G.f.: A(x) = 1 + 7*x + 91*x^2 + 1345*x^3 + 20995*x^4 + 337877*x^5 + ...,

%e where A(x)^2 = (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)).

%p 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

%t a[n_] := Hypergeometric2F1[-n, n + 1/2, 1, -4];

%t Table[a[n], {n, 0, 19}] (* _Peter Luschny_, Mar 16 2018 *)

%t 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 *)

%t a[n_] := Sum[(5^k Gamma[2 n + 1])/(Gamma[2 k + 1]*Gamma[n - k + 1]^2), {k, 0, n}];

%t Flatten[Table[a[n], {n, 0, 19}]] (* _Detlef Meya_, May 22 2024 *)

%o (PARI) /* From definition: */

%o {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)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* From a(n) = sqrt( A243945(2*n) ): */

%o {a(n)=sqrtint( sum(k=0, 2*n, binomial(2*k, k)^2*binomial(2*n+k, 2*n-k)) )}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n) = sum(k=0, n, 5^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))} \\ _Seiichi Manyama_, Aug 25 2020

%o (Python)

%o from math import comb

%o 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

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

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Aug 17 2014