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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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12
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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, 17383387729001, 32968493968795
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OFFSET
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0,4
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COMMENTS
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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, 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, Feb 06 2008
Also numerator of (1/Pi)*int(x^n*sqrt((1-x)/x), x=0..1). - Groux Roland, Mar 17 2011
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
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..500
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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, May 21 2008
# second Maple program:
a:= n-> abs(numer(binomial(1/2, n+1))): seq(a(n), n=0..50); # Alois P. Heinz, Apr 10 2009
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MATHEMATICA
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Table[Numerator[CatalanNumber[n]/2^(2n+1)], {n, 0, 30}] (* Harvey P. Dale, Jul 27 2011 *)
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PROG
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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 * A097038 A049114 A179189
Adjacent sequences: A098594 A098595 A098596 * 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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STATUS
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approved
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