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 A243943 #27 May 19 2023 15:19:00
%S A243943 1,25,1369,93025,6974881,553425625,45558768025,3848757330625,
%T A243943 331434586569025,28966516730025625,2561512789823546329,
%U A243943 228690489716580520225,20579914168308199841761,1864413002713001259355225,169871744046114667846619929,15554069096581207471331850625
%N A243943 a(n) = A006442(n)^2.
%C A243943 In general, we have the binomial identity:
%C A243943 if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k),
%C A243943 then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k),
%C A243943 where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2),
%C A243943 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 A243943 Note that the g.f. of A006442 is 1/sqrt(1 - 10*x + x^2).
%C A243943 Limit_{n -> oo} a(n+1)/a(n) = (5 + 2*sqrt(6))^2 = 49 + 20*sqrt(6).
%H A243943 Seiichi Manyama, <a href="/A243943/b243943.txt">Table of n, a(n) for n = 0..500</a>
%F A243943 G.f.: 1 / AGM(1-x, sqrt(1-98*x+x^2)). - _Paul D. Hanna_, Aug 30 2014
%F A243943 a(n) = Sum_{k=0..n} 6^k * C(2*k, k)^2 * C(n+k, n-k).
%F A243943 a(n)^(1/2) = Sum_{k=0..n} 2^k * C(2*k, k) * C(n+k, n-k).
%F A243943 a(n) ~ (5+2*sqrt(6))^(2*n+1) / (4*Pi*sqrt(6)*n). - _Vaclav Kotesovec_, Sep 28 2019
%e A243943 G.f.: A(x) = 1 + 9*x + 169*x^2 + 3969*x^3 + 103041*x^4 + 2832489*x^5 +...
%t A243943 Table[Sum[6^k * Binomial[2*k, k]^2 * Binomial[n+k, n-k], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 28 2019 *)
%t A243943 LegendreP[Range[0,40], 5]^2 (* _G. C. Greubel_, May 17 2023 *)
%o A243943 (PARI) {a(n) = sum(k=0, n, 6^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
%o A243943 for(n=0, 20, print1(a(n), ", "))
%o A243943 (PARI) {a(n) = sum(k=0, n, 2^k * binomial(2*k, k) * binomial(n+k, n-k) )^2}
%o A243943 for(n=0, 20, print1(a(n), ", "))
%o A243943 (PARI) /* Using AGM: */
%o A243943 {a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 10^2*x +x*O(x^n))), n)}
%o A243943 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 30 2014
%o A243943 (Magma) [Evaluate(LegendrePolynomial(n), 5)^2 : n in [0..40]]; // _G. C. Greubel_, May 17 2023
%o A243943 (SageMath) [gen_legendre_P(n,0,5)^2 for n in range(41)] # _G. C. Greubel_, May 17 2023
%Y A243943 Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), A243949 (m=1), this sequence (m=2), A243944 (m=3), A243007 (m=4).
%Y A243943 Cf. A006442.
%K A243943 nonn
%O A243943 0,2
%A A243943 _Paul D. Hanna_, Aug 17 2014