OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
W. Lang, On polynomials related to derivatives of the generating function of Catalan numbers, Fib. Quart. 40,4 (2002) 299-313; Eq.(31) with lambda=-1/2.
FORMULA
a(n) = (-1)^n*Sum_{k=0,..,n} (C(k)/(-1/2)^k) with C(k)=A000108(k) (Catalan).
a(n) = -a(n-1) + C(n)*2^n, n >= 0, a(-1) := 0, with C(n)=A000108(n).
G.f.: A(2*x)/(1+x), with A(x) g.f. of Catalan numbers A000108.
Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Dec 09 2013
a(n) ~ 2^(3*n+3)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 09 2013
MATHEMATICA
CoefficientList[Series[(1-Sqrt[1-8*x])/(4*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 09 2013 *)
Table[FullSimplify[2^(n+1)*(2*n+2)! * Hypergeometric2F1Regularized[1, n+3/2, n+3, -8]/(n+1)! + (-1)^n/2], {n, 0, 20}] (* Vaclav Kotesovec, Dec 09 2013 *)
Table[(-1)^n*Sum[(-2)^k * CatalanNumber[k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Jan 27 2017 *)
PROG
(Sage)
def A064306():
f, c, n = 1, 1, 1
while True:
yield f
n += 1
c = c * (8*n - 12) // n
f = c - f
a = A064306()
print([next(a) for _ in range(21)]) # Peter Luschny, Nov 30 2016
(PARI) for(n=0, 25, print1((-1)^n*sum(k=0, n, (-2)^k*binomial(2*k, k)/(k+1)), ", ")) \\ G. C. Greubel, Jan 27 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 13 2001
STATUS
approved