OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
FORMULA
a(n) = Sum_{k=0..n} C(k)*2^k, n >= 0, with C(n):=A000108(n).
G.f.: c(2*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
a(n) = Sum_{j=0..n} binomial(2*j,j)*2^j/(j+1). - Zerinvary Lajos, Oct 26 2006
Recurrence: (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(3*n+3)/(7*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012
MAPLE
a:=n->sum((binomial(2*j, j))*2^j/(j+1), j=0..n): seq(a(n), n=0..20); # Zerinvary Lajos, Oct 26 2006
MATHEMATICA
Table[Sum[Binomial[2*j, j]*2^j/(j+1), {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 19 2012 *)
PROG
(Sage)
def A112696():
f, c, n = 1, 1, 1
while True:
yield f
n += 1
c = c * (8*n - 12) // n
f += c
a = A112696()
print([next(a) for _ in range(21)]) # Peter Luschny, Nov 30 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Oct 31 2005
STATUS
approved