OFFSET
0,4
COMMENTS
Radius of convergence of g.f. is r = 1/2.
More generally, given
A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
then A(x) = 1/sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t > 0.
a(n) is the number of (2k-1)-element subsets of {1, 2, ..., n+1} whose k-th smallest (i.e., k-th largest) element equals 2k-1. - Darij Grinberg, Oct 09 2019
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = 1 / sqrt( (1 - x + 2*x^2)^2 - 4*x^2 ).
G.f.: A(x) = 1 / sqrt( (1-x)*(1-2*x)*(1+x+2*x^2) ).
G.f. satisfies: A(x) = (1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)) / (1-x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.
a(n) ~ 2^n/sqrt(Pi*n). - Vaclav Kotesovec, Sep 16 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k, k) * binomial(n-2*k, k). - Darij Grinberg, Oct 09 2019
a(n) = hypergeom([1/2,(1-n)/3,(2-n)/3, -n/3], [1, (1-n)/2, -n/2], -27)) for n >= 4. - Peter Luschny, Oct 09 2019
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 15*x^6 + 29*x^7 + 49*x^8 + ...
where the g.f. equals the series:
A(x) = 1 +
x*((1-x) + x) +
x^2*((1-x)^2 + 2^2*x*(1-x) + x^2) +
x^3*((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3) +
x^4*((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4) +
x^5*((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5) + ...
MAPLE
a := n -> `if`(n < 4, [1, 1, 1, 3][n+1], hypergeom([1/2, (1-n)/3, (2-n)/3, -n/3], [1, (1-n)/2, -n/2], -27)):
seq(simplify(a(n)), n=0..36); # Peter Luschny, Oct 09 2019
MATHEMATICA
CoefficientList[Series[1/Sqrt[(1-x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2013 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(1-x)^(m-k) + x*O(x^n))), n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) a(n)={sum(k=0, n\2, binomial(2*k, k) * binomial(n-2*k, k))} \\ Andrew Howroyd, Oct 09 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 09 2012
STATUS
approved