A243945 - OEIS

%I #47 Sep 08 2022 08:46:08

%S 1,5,49,605,8281,120125,1809025,27966125,440790025,7051890125,

%T 114160867129,1865975723045,30743797894681,509948702030045,

%U 8507207970913729,142626515754330125,2401552098016698025,40591712338241826125,688413807606268692025,11710401759994742685125

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

%C The g.f.s formed from a(2*n)^(1/2) and (a(2*n+1)/5)^(1/2) are:

%C A243946: sqrt( (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)) );

%C A243947: sqrt( (1+x - sqrt(1-18*x+x^2)) / (10*x*(1-18*x+x^2)) ).

%C Lim_{n->infinity} a(n+1)/a(n) = 9 + 4*sqrt(5).

%C Diagonal of rational function 1/(1 - (x + y + x*z + y*z + x*y*z)). - _Gheorghe Coserea_, Aug 24 2018

%H Seiichi Manyama, <a href="/A243945/b243945.txt">Table of n, a(n) for n = 0..800</a>

%H A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

%F G.f.: Sum_{n>=0} binomial(2*n, n)^2 * x^n / (1-x)^(2*n+1).

%F G.f.: 1 / AGM(1-x, sqrt(1-18*x+x^2)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

%F a(2*n) = A243946(n)^2.

%F a(2*n+1) = 5 * A243947(n)^2.

%F Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(19*n^2 - 38*n + 14)*a(n-1) - (2*n-3)*(19*n^2 - 38*n + 14)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - _Vaclav Kotesovec_, Aug 18 2014

%F a(n) ~ (2+sqrt(5)) * (9+4*sqrt(5))^n / (4*Pi*n). - _Vaclav Kotesovec_, Aug 18 2014. Equivalently, a(n) ~ phi^(6*n + 3) / (4*Pi*n), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 08 2021

%F a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -4). - _Peter Luschny_, Mar 14 2018

%F G.f. y=A(x) satisfies: 0 = x*(x^2 - 1)*(x^2 - 18*x + 1)*y'' + (3*x^4 - 34*x^3 - 38*x^2 + 38*x - 1)*y' + (x^3 - 3*x^2 - 19*x + 5)*y. - _Gheorghe Coserea_, Aug 29 2018

%F From _Peter Bala_, Feb 07 2022: (Start)

%F a(n) = P(n,sqrt(5))^2, where P(n,x) denotes the n-th Legendre polynomial.

%F The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for all primes p and positive integers n and k. (End)

%e G.f.: A(x) = 1 + 5*x + 49*x^2 + 605*x^3 + 8281*x^4 + 120125*x^5 + ... where

%e A(x) = 1/(1-x) + 2^2*x/(1-x)^3 + 6^2*x^2/(1-x)^5 + 20^2*x^3/(1-x)^7 + 70^2*x^4/(1-x)^9 + 252^2*x^5/(1-x)^11 + 924^2*x^6/(1-x)^13 + ... + A000984(n)^2*x^n/(1-x)^(2*n+1) + ...

%t Table[Sum[Binomial[2*k, k]^2 * Binomial[n + k, n - k],{k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 18 2014 *)

%t a[n_] := HypergeometricPFQ[{1/2, -n, n + 1}, {1, 1}, -4];

%t Table[a[n], {n, 0, 19}] (* _Peter Luschny_, Mar 14 2018 *)

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

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

%o (PARI) {a(n)=local(A=1); A=sum(m=0, n, binomial(2*m, m)^2 * x^m/(1-x +x*O(x^n))^(2*m+1)); polcoeff(A, n)}

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

%o (PARI) {a(n)=polcoeff( 1 / agm(1-x, sqrt((1-x)^2 - 16*x +x*O(x^n))), n)}

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

%o (Magma) &cat[ [&+[ Binomial(2*k, k)^2 * Binomial(n+k, n-k): k in [0..n]]]: n in [0..30]]; // _Vincenzo Librandi_, Aug 25 2018

%Y Cf. A243946, A243947, A245925.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 17 2014