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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. 13

%I

%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

%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>

%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 *)

%o (PARI) a(n)=if(n<0,0,numerator(polcoeff(1/(1+sqrt(1-x+x^n*O(x))),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

%O 0,4

%A _Michael Somos_, Sep 15 2004

%E Edited by _Ralf Stephan_, Dec 28 2004

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Last modified October 23 14:06 EDT 2018. Contains 316528 sequences. (Running on oeis4.)