A098597 Numerator of Catalan(n)/2^(2n+1). Also, numerators of (2n-1)!!/(n+1)!. Odd part of the n-th Catalan number.

1, 1, 1, 5, 7, 21, 33, 429, 715, 2431, 4199, 29393, 52003, 185725, 334305, 9694845, 17678835, 64822395, 119409675, 883631595, 1641030105, 6116566755, 11435320455, 171529806825, 322476036831, 1215486600363, 2295919134019, 17383387729001, 32968493968795, 125280277081421

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

Formula

Numerators of g.f.: 1/(1 + sqrt(1-x)).

a(n) = A000108(n) / 2^A048881(n).

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

Cf. Equals A000265(A000108(n)).

Essentially the absolute values of A002596. Cf. A000108, A001795.

Sequence in context: A027152 A076197 A002596 * A097038 A049114 A179189

Adjacent sequences: A098594 A098595 A098596 * A098598 A098599 A098600

Keyword

nonn,frac,changed

Author

Michael Somos, Sep 15 2004

Extensions

Edited by Ralf Stephan, Dec 28 2004

Status

approved