OFFSET
0,4
COMMENTS
Also numerators of g.f. c(x/2) = (1-sqrt(1-2x))/x where c(x) = g.f. of A000108. - Paul Barry, Sep 04 2007
Also numerator of x(n)=Sum(x(k)*x(n-k-1):0<=k<n), x(0)=1/2: x(n)=a(n)/A086117(n). - Reinhard Zumkeller, Feb 06 2008
Also numerator of (1/Pi)*int(x^n*sqrt((1-x)/x), x=0..1). - Groux Roland, Mar 17 2011
The negative of this sequence appears in the A-sequence of the Riordan triangle A084930 as numerators 4, -2, -seq(a(n-1), n >= 2). The denominators look like 1, seq(A120777(n-1), n >= 1). - Wolfdieter Lang, Aug 04 2014
The series of a(n)/A046161(n+1) is absolutely convergent to 1. - Ralf Steiner, Feb 09 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
T. Copeland, Addendum to Elliptic Lie Triad
EXAMPLE
1/(1 + sqrt(1-x)) = 1/2 + 1/8*x + 1/16*x^2 + 5/128*x^3 + 7/256*x^4 + ...
MAPLE
a:= n-> abs(numer(binomial(1/2, n+1))): seq(a(n), n=0..50); # Alois P. Heinz, Apr 10 2009
MATHEMATICA
Table[Numerator[CatalanNumber[n]/2^(2n+1)], {n, 0, 30}] (* Harvey P. Dale, Jul 27 2011 *)
A098597[n_] := With[{c = CatalanNumber[n]}, c / 2^IntegerExponent[c, 2]];
Table[A098597[n], {n, 0, 29}] (* Peter Luschny, Apr 16 2024 *)
PROG
(PARI) {a(n) = if( n < 0, 0, numerator(polcoeff(1 / (1 + sqrt(1 - x + x * O(x^n))), n)))};
(Magma) [Numerator(Catalan(n)/2^(2*n+1)):n in [0..30]]; // Vincenzo Librandi, Jan 14 2016
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Michael Somos, Sep 15 2004
EXTENSIONS
Edited by Ralf Stephan, Dec 28 2004
STATUS
approved