A064306 Convolution of A052701 (Catalan numbers multiplied by powers of 2) with powers of -1.

1, 1, 7, 33, 191, 1153, 7295, 47617, 318463, 2170881, 15028223, 105365505, 746651647, 5339185153, 38478839807, 279201841153, 2037998419967, 14954803494913, 110255315877887, 816299567480833, 6066679566041087

Offset: 0

Author

Wolfdieter Lang, Sep 13 2001

Keywords

• nonn
• easy

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.

Programs

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 *)

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

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

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