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a(n) = A084769(n)^2.
5

%I #23 May 20 2023 15:20:36

%S 1,81,14641,3272481,806616801,210358905201,56912554609681,

%T 15800522430616641,4471485120646226881,1284238494711502355601,

%U 373195323236525968732401,109489964937514282794301281,32378265673661271315300820641,9639042117142706280223219663281

%N a(n) = A084769(n)^2.

%C In general, we have the binomial identity:

%C if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k),

%C then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k),

%C where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2),

%C 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.

%C Note that the g.f. of A084769 is 1/sqrt(1 - 18*x + x^2).

%C Limit_{n -> oo} a(n+1)/a(n) = (9 + 4*sqrt(5))^2 = 161 + 72*sqrt(5).

%H Vincenzo Librandi, <a href="/A243007/b243007.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: 1 / AGM(1-x, sqrt(1- 322*x + x^2)). - _Paul D. Hanna_, Aug 30 2014

%F a(n) = Sum_{k=0..n} 20^k * C(2*k, k)^2 * C(n+k, n-k).

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

%F a(n) ~ (2 + sqrt(5))^(4*n+2) / (8*sqrt(5)*Pi*n). - _Vaclav Kotesovec_, Sep 28 2019

%e G.f.: A(x) = 1 + 81*x + 14641*x^2 + 3272481*x^3 + 806616801*x^4 +...

%t Table[SeriesCoefficient[1/Sqrt[1 -18x +x^2], {x,0,n}], {n,0,20}]^2 (* _Vincenzo Librandi_, Feb 14 2018 *)

%t LegendreP[Range[0,30], 9]^2 (* _G. C. Greubel_, May 17 2023 *)

%o (PARI) {a(n) = sum(k=0, n, 20^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}

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

%o (PARI) {a(n) = sum(k=0, n, 4^k * binomial(2*k, k) * binomial(n+k, n-k) )^2}

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

%o {a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 18^2*x +x*O(x^n))), n)}

%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 30 2014

%o (Magma) [Evaluate(LegendrePolynomial(n),9)^2 : n in [0..30]]; // _G. C. Greubel_, May 17 2023

%o (SageMath) [gen_legendre_P(n,0,9)^2 for n in range(41)] # _G. C. Greubel_, May 17 2023

%Y Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), A243949 (m=1), A243943 (m=2), A243944 (m=3), this sequence (m=4).

%Y Cf. A084769.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 18 2014