A243944 - OEIS

%I #26 May 20 2023 15:20:47

%S 1,49,5329,717409,106523041,16735820689,2727812288881,456250924320961,

%T 77788137919752001,13459803510972477169,2356471368269511061009,

%U 416518496068852312607521,74207592486779379593752801,13309569813247406938272432721,2400816685486139045360488325809

%N a(n) = A084768(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*(t+1))^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 A084768 is 1/sqrt(1 - 14*x + x^2).

%C Limit a(n+1)/a(n) = (7 + 4*sqrt(3))^2 = 97 + 56*sqrt(3).

%H Seiichi Manyama, <a href="/A243944/b243944.txt">Table of n, a(n) for n = 0..438</a>

%F G.f.: 1 / AGM(1-x, sqrt(1-194*x+x^2)).  - _Paul D. Hanna_, Aug 30 2014

%F a(n) = Sum_{k=0..n} 12^k * C(2*k, k)^2 * C(n+k, n-k).

%F a(n)^(1/2) = Sum_{k=0..n} 3^k * C(2*k, k) * C(n+k, n-k).

%F a(n) ~ (1+sqrt(3))^(8*n+4) / (sqrt(3) * Pi * n * 2^(4*n+5)). - _Vaclav Kotesovec_, Sep 28 2019

%F a(n) = (LegendreP(n, 7))^2. - _G. C. Greubel_, May 17 2023

%e G.f.: A(x) = 1 + 49*x + 5329*x^2 + 717409*x^3 + 106523041*x^4 +...

%t 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 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 LegendreP[Range[0, 40], 7]^2 (* _G. C. Greubel_, May 17 2023 *)

%o (PARI) {a(n) = sum(k=0, n, 12^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, 3^k * binomial(2*k, k) * binomial(n+k, n-k) )^2}

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

%o (PARI) /* Using AGM: */

%o {a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 14^2*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),7)^2 : n in [0..40]]; // _G. C. Greubel_, May 17 2023

%o (SageMath) [gen_legendre_P(n,0,7)^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), this sequence (m=3), A243007 (m=4).

%Y Cf. A084768.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 18 2014