OFFSET
0,2
COMMENTS
a(n) equals the number of sequences (x(1),...,x(n)) of n numbers from {0,1,...,n} such that x(i+1) <= x(i)+1 for i=1,...,n-1 and x(1) <= x(n)+1. This is proved in a linked pdf, as well as another interpretation as the downward closed subsets of a certain poset. - Clayton Thomas, Jul 16 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Clayton Thomas, A note on the cylinder poset
FORMULA
G.f.: A(x) = 1/sqrt(1-4*x) + 4*x^2/(1+sqrt(1-4*x))^2/(1-4*x)^(3/2).
a(n) = (n+3)*C(2*n-1,n) - 2^(2*n-1), a(n) ~ 2^(2*n - 1) * sqrt(n) / sqrt(Pi). - Vaclav Kotesovec, Oct 28 2012
Conjecture: a(n) = 2^(2*n)*(Sum_{j=1..n+2-floor((n+3)/2)} (cos(j*Pi/(n+3)))^(2*n)). - L. Edson Jeffery, Nov 23 2013
MATHEMATICA
Flatten[{1, Table[(n+3)*Binomial[2*n-1, n]-2^(2*n-1), {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 28 2012 *)
PROG
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(1/sqrt(1-4*X)+4*X^2/(1+sqrt(1-4*X))^2/(1-4*X)^(3/2), n, x)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2006
STATUS
approved