%I A002596 M3768 N1538
%S A002596 1,1,1,1,5,7,21,33,429,715,2431,4199,29393,52003,185725,334305,
%T A002596 9694845,17678835,64822395,119409675,883631595,1641030105,6116566755,
%U A002596 11435320455,171529806825,322476036831,1215486600363,2295919134019
%V A002596 1,1,-1,1,-5,7,-21,33,-429,715,-2431,4199,-29393,52003,-185725,334305,
%W A002596 -9694845,17678835,-64822395,119409675,-883631595,1641030105,-6116566755,
%X A002596 11435320455,-171529806825,322476036831,-1215486600363,2295919134019
%N A002596 Numerators in expansion of sqrt(1+x). Absolute values give numerators 
               in expansion of (1-x)^(1/2).
%C A002596 Also, absolute values are numerators of (2n-3)!!/n! or the odd part of 
               the (n-1)th Catalan number.
%D A002596 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002596 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002596 B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 
               1, p. 513, Eq. (7.281).
%H A002596 T. D. Noe, <a href="b002596.txt">Table of n, a(n) for n=0..200</a>
%H A002596 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LegendrePolynomial.html">Legendre Polynomial</a>
%F A002596 a(n+2) = C(n+1)/2^k(n+1), n >= 0; C(n)= A000108(n)(Catalan), k(n)= A048881(n).
%F A002596 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 
               2009: (Start)
%F A002596 a(n) = (-1)^n*numer((1/(1-2*n))*binomial(2*n,n)/(4^n))
%F A002596 (1+x)^(1/2) = sum((1/(1-2*n))*binomial(2*n,n)/(4^n)*(-x)^n, n=0..infinity)
%F A002596 (1-x)^(1/2) = sum((1/(1-2*n))*binomial(2*n,n)/(4^n)*(x)^n, n=0..infinity)
%F A002596 (End)
%e A002596 sqrt(1+x) = 1+1/2*x-1/8*x^2+1/16*x^3-5/128*x^4+7/256*x^5-21/1024*x^6+33/
               2048*x^7+...
%t A002596 InverseSeries[Series[2^p*y-y^2/2^q, {y, 0, 24}], x] (* p, q positive 
               integers, then a(n)=numerator(y(n)) *) - Len Smiley, Apr 13 2000
%Y A002596 Denominators are A046161.
%Y A002596 Cf. A001795.
%Y A002596 Equals A000265(A000108(n-1)), n>0.
%Y A002596 Absolute values are essentially A098597.
%Y A002596 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 
               2009: (Start)
%Y A002596 Cf. A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(`5/2)].
%Y A002596 Cf. A001803 [1-x)^(-3/2)]
%Y A002596 A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/
               2) for all values of n.
%Y A002596 (End)
%Y A002596 Sequence in context: A057424 A027152 A076197 this_sequence A098597 A097038 
               A049114
%Y A002596 Adjacent sequences: A002593 A002594 A002595 this_sequence A002597 A002598 
               A002599
%K A002596 easy,nice,frac,sign
%O A002596 0,5
%A A002596 N. J. A. Sloane (njas(AT)research.att.com).
%E A002596 Minor correction to definition from Johannes W. Meijer, Jun 05 2009