|
%I #48 Apr 16 2024 12:05:29
%S 1,1,1,5,7,21,33,429,715,2431,4199,29393,52003,185725,334305,9694845,
%T 17678835,64822395,119409675,883631595,1641030105,6116566755,
%U 11435320455,171529806825,322476036831,1215486600363,2295919134019,17383387729001,32968493968795,125280277081421
%N Numerator of Catalan(n)/2^(2n+1). Also, numerators of (2n-1)!!/(n+1)!. Odd part of the n-th Catalan number.
%C 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
%C 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
%C Also numerator of (1/Pi)*int(x^n*sqrt((1-x)/x), x=0..1). - _Groux Roland_, Mar 17 2011
%C 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
%C The series of a(n)/A046161(n+1) is absolutely convergent to 1. - _Ralf Steiner_, Feb 09 2017
%H Alois P. Heinz, <a href="/A098597/b098597.txt">Table of n, a(n) for n = 0..500</a>
%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>
%F Numerators of g.f.: 1/(1 + sqrt(1-x)).
%F a(n) = A000108(n) / 2^A048881(n).
%e 1/(1 + sqrt(1-x)) = 1/2 + 1/8*x + 1/16*x^2 + 5/128*x^3 + 7/256*x^4 + ...
%p a:= n-> abs(numer(binomial(1/2, n+1))): seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 10 2009
%t Table[Numerator[CatalanNumber[n]/2^(2n+1)],{n,0,30}] (* _Harvey P. Dale_, Jul 27 2011 *)
%t A098597[n_] := With[{c = CatalanNumber[n]}, c / 2^IntegerExponent[c, 2]];
%t Table[A098597[n], {n, 0, 29}] (* _Peter Luschny_, Apr 16 2024 *)
%o (PARI) {a(n) = if( n < 0, 0, numerator(polcoeff(1 / (1 + sqrt(1 - x + x * O(x^n))), n)))};
%o (Magma) [Numerator(Catalan(n)/2^(2*n+1)):n in [0..30]]; // _Vincenzo Librandi_, Jan 14 2016
%Y Cf. Equals A000265(A000108(n)).
%Y Essentially the absolute values of A002596. Cf. A000108, A001795.
%K nonn,frac,changed
%O 0,4
%A _Michael Somos_, Sep 15 2004
%E Edited by _Ralf Stephan_, Dec 28 2004
|
