OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1658
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
FORMULA
a(n) = binomial(2*n,n) * (2*n + 1 - hypergeom([1,-n], [1/2-n], 1/4))/3.
a(n+1) - a(n) = A002457(n) = (2*n+1)!/n!^2.
Recurrence: (5*n + 2) * a(n) = (4*n + 2) * a(n-1) + n * a(n+1).
a(n) ~ sqrt(n) * 2^(2*n+1) / (3*sqrt(Pi)). - Vaclav Kotesovec, Jan 29 2019
G.f.: x/(1-x) * (1-4*x)^(-3/2). - Seiichi Manyama, Jan 29 2019
MAPLE
a:=n->sqrt(-1/27)-((n+1)/2)*binomial(2*(n+1), n+1)*hypergeom([1, n+3/2], [n+1], 4):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Oct 03 2016
MATHEMATICA
Table[Binomial[2 n, n] (2 n + 1 - Hypergeometric2F1[1, -n, 1/2 - n, 1/4])/3, {n, 0, 30}]
PROG
(PARI) {a(n) = sum(k=0, n, k*binomial(2*k, k))/2} \\ Seiichi Manyama, Jan 29 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Reshetnikov, Oct 02 2016
STATUS
approved