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 A098597 Numerator of Catalan(n)/2^(2n+1). Also, numerators of (2n-1)!!/(n+1)!. Odd part of the n-th Catalan number. 14
 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 (list; graph; refs; listen; history; text; internal format)
 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= 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 AUTHOR Michael Somos, Sep 15 2004 EXTENSIONS Edited by Ralf Stephan, Dec 28 2004 STATUS approved

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Last modified August 8 12:20 EDT 2024. Contains 375021 sequences. (Running on oeis4.)