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Search: id:A098597
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| A098597 |
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Numerator of Catalan(n)/2^(2n+1). Also, numerators of (2n-1)!!/(n+1)!. Odd part of the n-th Catalan number. |
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+0
5
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| 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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0,4
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COMMENT
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Also numerators of g.f. c(x/2)=(1-sqrt(1-2x))/x where c(x)=g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), Sep 04 2007
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 (reinhard.zumkeller(AT)gmail.com), Feb 06 2008
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FORMULA
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Numerators of g.f.: 1/(1+sqrt(1-x)).
a(n) = A000108(n) / 2^A048881(n).
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EXAMPLE
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1/(1+sqrt(1-x)) = 1/2 + 1/8*x + 1/16*x^2 + 5/128*x^3 + 7/256*x^4 +...
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MAPLE
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Z[0]:=0: for k to 30 do Z[k]:=simplify(1/(2-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(numer(coeff(gser, z, n)), n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 21 2008
a:= n-> abs (numer (binomial (1/2, n+1))): seq (a(n), n=0..50); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Apr 10 2009]
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PROGRAM
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(PARI) a(n)=if(n<0, 0, numerator(polcoeff(1/(1+sqrt(1-x+x^n*O(x))), n)))
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CROSSREFS
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Cf. Equals A000265(A000108(n)).
Essentially the absolute values of A002596. Cf. A000108, A001795.
Sequence in context: A027152 A076197 A002596 this_sequence A097038 A049114 A030735
Adjacent sequences: A098594 A098595 A098596 this_sequence A098598 A098599 A098600
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KEYWORD
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nonn,frac
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AUTHOR
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Michael Somos, Sep 15 2004
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EXTENSIONS
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Edited by Ralf Stephan, Dec 28 2004
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