This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A243007 #23 May 20 2023 15:20:36
%S A243007 1,81,14641,3272481,806616801,210358905201,56912554609681,
%T A243007 15800522430616641,4471485120646226881,1284238494711502355601,
%U A243007 373195323236525968732401,109489964937514282794301281,32378265673661271315300820641,9639042117142706280223219663281
%N A243007 a(n) = A084769(n)^2.
%C A243007 In general, we have the binomial identity:
%C A243007 if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k),
%C A243007 then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k),
%C A243007 where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2),
%C A243007 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 A243007 Note that the g.f. of A084769 is 1/sqrt(1 - 18*x + x^2).
%C A243007 Limit_{n -> oo} a(n+1)/a(n) = (9 + 4*sqrt(5))^2 = 161 + 72*sqrt(5).
%H A243007 Vincenzo Librandi, <a href="/A243007/b243007.txt">Table of n, a(n) for n = 0..100</a>
%F A243007 G.f.: 1 / AGM(1-x, sqrt(1- 322*x + x^2)). - _Paul D. Hanna_, Aug 30 2014
%F A243007 a(n) = Sum_{k=0..n} 20^k * C(2*k, k)^2 * C(n+k, n-k).
%F A243007 a(n)^(1/2) = Sum_{k=0..n} 4^k * C(2*k, k) * C(n+k, n-k).
%F A243007 a(n) ~ (2 + sqrt(5))^(4*n+2) / (8*sqrt(5)*Pi*n). - _Vaclav Kotesovec_, Sep 28 2019
%e A243007 G.f.: A(x) = 1 + 81*x + 14641*x^2 + 3272481*x^3 + 806616801*x^4 +...
%t A243007 Table[SeriesCoefficient[1/Sqrt[1 -18x +x^2], {x,0,n}], {n,0,20}]^2 (* _Vincenzo Librandi_, Feb 14 2018 *)
%t A243007 LegendreP[Range[0,30], 9]^2 (* _G. C. Greubel_, May 17 2023 *)
%o A243007 (PARI) {a(n) = sum(k=0, n, 20^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
%o A243007 for(n=0, 20, print1(a(n), ", "))
%o A243007 (PARI) {a(n) = sum(k=0, n, 4^k * binomial(2*k, k) * binomial(n+k, n-k) )^2}
%o A243007 for(n=0, 20, print1(a(n), ", "))
%o A243007 {a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 18^2*x +x*O(x^n))), n)}
%o A243007 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 30 2014
%o A243007 (Magma) [Evaluate(LegendrePolynomial(n),9)^2 : n in [0..30]]; // _G. C. Greubel_, May 17 2023
%o A243007 (SageMath) [gen_legendre_P(n,0,9)^2 for n in range(41)] # _G. C. Greubel_, May 17 2023
%Y A243007 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 A243007 Cf. A084769.
%K A243007 nonn
%O A243007 0,2
%A A243007 _Paul D. Hanna_, Aug 18 2014