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 A243944 #26 May 20 2023 15:20:47
%S A243944 1,49,5329,717409,106523041,16735820689,2727812288881,456250924320961,
%T A243944 77788137919752001,13459803510972477169,2356471368269511061009,
%U A243944 416518496068852312607521,74207592486779379593752801,13309569813247406938272432721,2400816685486139045360488325809
%N A243944 a(n) = A084768(n)^2.
%C A243944 In general, we have the binomial identity:
%C A243944 if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k),
%C A243944 then b(n)^2 = Sum_{k=0..n} (t*(t+1))^k * C(2*k, k)^2 * C(n+k, n-k),
%C A243944 where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2),
%C A243944 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 A243944 Note that the g.f. of A084768 is 1/sqrt(1 - 14*x + x^2).
%C A243944 Limit a(n+1)/a(n) = (7 + 4*sqrt(3))^2 = 97 + 56*sqrt(3).
%H A243944 Seiichi Manyama, <a href="/A243944/b243944.txt">Table of n, a(n) for n = 0..438</a>
%F A243944 G.f.: 1 / AGM(1-x, sqrt(1-194*x+x^2)). - _Paul D. Hanna_, Aug 30 2014
%F A243944 a(n) = Sum_{k=0..n} 12^k * C(2*k, k)^2 * C(n+k, n-k).
%F A243944 a(n)^(1/2) = Sum_{k=0..n} 3^k * C(2*k, k) * C(n+k, n-k).
%F A243944 a(n) ~ (1+sqrt(3))^(8*n+4) / (sqrt(3) * Pi * n * 2^(4*n+5)). - _Vaclav Kotesovec_, Sep 28 2019
%F A243944 a(n) = (LegendreP(n, 7))^2. - _G. C. Greubel_, May 17 2023
%e A243944 G.f.: A(x) = 1 + 49*x + 5329*x^2 + 717409*x^3 + 106523041*x^4 +...
%t A243944 Table[Sum[12^k * Binomial[2*k, k]^2 * Binomial[n+k, n-k], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 28 2019 *)
%t A243944 CoefficientList[Series[2*EllipticK[1 - (1-x)^2/(1 - 194*x + x^2)] / (Pi*Sqrt[1 - 194*x + x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 28 2019 *)
%t A243944 LegendreP[Range[0, 40], 7]^2 (* _G. C. Greubel_, May 17 2023 *)
%o A243944 (PARI) {a(n) = sum(k=0, n, 12^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
%o A243944 for(n=0, 20, print1(a(n), ", "))
%o A243944 (PARI) {a(n) = sum(k=0, n, 3^k * binomial(2*k, k) * binomial(n+k, n-k) )^2}
%o A243944 for(n=0, 20, print1(a(n), ", "))
%o A243944 (PARI) /* Using AGM: */
%o A243944 {a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 14^2*x +x*O(x^n))), n)}
%o A243944 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 30 2014
%o A243944 (Magma) [Evaluate(LegendrePolynomial(n),7)^2 : n in [0..40]]; // _G. C. Greubel_, May 17 2023
%o A243944 (SageMath) [gen_legendre_P(n,0,7)^2 for n in range(41)] # _G. C. Greubel_, May 17 2023
%Y A243944 Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), A243949 (m=1), A243943 (m=2), this sequence (m=3), A243007 (m=4).
%Y A243944 Cf. A084768.
%K A243944 nonn
%O A243944 0,2
%A A243944 _Paul D. Hanna_, Aug 18 2014