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Search: id:A002596
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| A002596 |
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Numerators in expansion of sqrt(1+x). Absolute values give numerators in expansion of (1-x)^(1/2).
(Formerly M3768 N1538)
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+0
9
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| 1, 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
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Also, absolute values are numerators of (2n-3)!!/n! or the odd part of the (n-1)th Catalan number.
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REFERENCES
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B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 513, Eq. (7.281).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Eric Weisstein's World of Mathematics, Legendre Polynomial
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FORMULA
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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)
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EXAMPLE
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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+...
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MATHEMATICA
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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
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CROSSREFS
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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)
Adjacent sequences: A002593 A002594 A002595 this_sequence A002597 A002598 A002599
Sequence in context: A057424 A027152 A076197 this_sequence A098597 A097038 A049114
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KEYWORD
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easy,nice,frac,sign,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Minor correction to definition from Johannes W. Meijer, Jun 05 2009
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