E.g.f.: exp(x)*BesselI(0, 2*i*x), i=sqrt(-1);
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*(-1)^k;
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*(-1)^k;
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, k/2)*cos(pi*k/2).
D-finite with recurrence: a(0)=a(1)=1, a(n) = ((2n-1)a(n-1)-5(n-1)a(n-2))/n. -
T. D. Noe, Oct 19 2005
a(n) = hypergeom([-n/2, 1/2-n/2], [1], -4). -
Peter Luschny, Sep 18 2014
a(n) ~ 5^(n/2 + 1/4) * cos((Pi*n - arctan(1/2) - n*arctan(4/3))/2) / sqrt(Pi*n). -
Vaclav Kotesovec, Oct 31 2017
a(n) = (sqrt(5))^n*P(n,1/sqrt(5)), where P(n,x) is the Legendre polynomial of degree n. Note the general result (sqrt(4*m+1))^n*P(n, 1/sqrt(4*m+1)) = Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)(-m)^k due to Catalan. -
Peter Bala, Mar 18 2018
G.f.: 1/(1 - x + 2*x^2/(1 - x + x^2/(1 - x + x^2/(1 - x + x^2/(1 - ...))))), a continued fraction. -
Ilya Gutkovskiy, Nov 19 2021
G.f.: A(x) = Sum_{n >= 0} (-1)^n*binomial(2*n,n)*x^(2*n)/(1 - x)^(2*n+1).
a(n)^2 = Sum_{k = 0..n} (-1)^k*5^(n-k)*binomial(2*k,k)*binomial(n,k)* binomial(n+k,k).
Sum_{n >= 0} (-1)^n*binomial(2*n,n)^2 * x^n/(1 - 5*x)^(2*n+1) = 1 + x + x^2 + 25*x^3 + 25*x^4 + 121*x^5 + ... is the g.f. of a(n)^2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k. (End)
