0,5

Also, absolute values are numerators of (2n-3)!!/n! or the odd part of the (n-1)th Catalan number.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 513, Eq. (7.281).

T. D. Noe, Table of n, a(n) for n=0..200

Eric Weisstein's World of Mathematics, Legendre Polynomial

a(n+2) = C(n+1)/2^k(n+1), n >= 0; C(n)= A000108(n)(Catalan), k(n)= A048881(n).

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)

a(n) = (-1)^n*numer((1/(1-2*n))*binomial(2*n,n)/(4^n))

(1+x)^(1/2) = sum((1/(1-2*n))*binomial(2*n,n)/(4^n)*(-x)^n, n=0..infinity)

(1-x)^(1/2) = sum((1/(1-2*n))*binomial(2*n,n)/(4^n)*(x)^n, n=0..infinity)

(End)

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+...

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

Denominators are A046161.

Cf. A001795.

Equals A000265(A000108(n-1)), n>0.

Absolute values are essentially A098597.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)

Cf. A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(`5/2)].

Cf. A001803 [1-x)^(-3/2)]

A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n.

(End)

Sequence in context: A057424 A027152 A076197 this_sequence A098597 A097038 A049114

Adjacent sequences: A002593 A002594 A002595 this_sequence A002597 A002598 A002599

easy,nice,frac,sign

N. J. A. Sloane (njas(AT)research.att.com).

Minor correction to definition from Johannes W. Meijer, Jun 05 2009