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Search: id:A068875
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| A068875 |
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Expansion of (1+x*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108. |
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7
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| 1, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024, 1485800, 5348880, 19389690, 70715340, 259289580, 955277400, 3534526380, 13128240840, 48932534040, 182965127280, 686119227300, 2579808294648, 9723892802904
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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A Catalan transform of A040000 under the mapping g(x)->g(xc(x)). A040000 can be retrieved using the mapping g(x)->g(x(1-x)). A040000(n)=sum{k=0..floor(n/2), C(n-k,k)(-1)^k*A068875(n-k)}. A068875 and A040000 may be described as a Catalan pair. - Paul Barry (pbarry(AT)wit.ie), Nov 14 2004
a(n) = number of Dyck (n+1)-paths all of whose nonterminal descents to ground level are of odd length. For example, a(2) counts UUUDDD, UUDUDD, UDUUDD, UDUDUD. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005
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REFERENCES
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Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
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FORMULA
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Apart from initial term, twice Catalan numbers.
G.f.: (1+xc(x))/(1-xc(x)), where c(x) is the g.f. of A000108; a(n)=C(n)(2-0^n); C(n) as in A000108; a(n)=sum{j=0..n, sum{k=0..n, C(2n, n-k)((2k+1)/(n+k+1))C(k, j)(-1)^(j-k)*(2-0^j)}}. - Paul Barry (pbarry(AT)wit.ie), Nov 14 2004
Assuming offset 1, then series reversion of g.f. A(x) is -A(-x). - Michael Somos Aug 17 2005
Assuming offset 2, then A(x) satisfies A(x - x^2) = x^2 - x^4 and so A(x)=C(x)^2-C(x)^4, A(A(x))=C(x)^4-C(x)^8, A(A(A(x)))=C(x)^8-C(x)^16, etc., where C(x)=(1-sqrt(1-4*x))/2 = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 +... - Paul D. Hanna (pauldhanna(AT)juno.com), May 16 2008
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MAPLE
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Z:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*Z)/Z: Gser:=series(-G, x=0, 30): seq(coeff(Gser, x, n), n=2..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 23 2006
Z:=-(1-z-sqrt(1-z))/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser*4^n, z, n), n=1..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2007
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PROGRAM
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(PARI) a(n)=if(n<1, n==0, 2*binomial(2*n, n)/(n+1)) /* Michael Somos Aug 17 2005 */
(PARI) a(n)=if(n<1, n==0, polcoeff(4/(1+sqrt(1-4*x+x*O(x^n))), n)) /* Michael Somos Aug 17 2005 */
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CROSSREFS
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Cf. A068875, A068875.
Adjacent sequences: A068872 A068873 A068874 this_sequence A068876 A068877 A068878
Sequence in context: A148110 A149823 A149824 this_sequence A135336 A149825 A149826
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Jun 06 2002
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