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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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8
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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; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
| 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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LINKS
| Guo-Niu Han, Enumeration of Standard Puzzles
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FORMULA
| 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
Apart from initial term, INVERTi transform of A000984(n+1)=binomial(2*n+2,n+1), also, for n>=1, a(n)=(1/Pi)*int(x^(n-1)*sqrt(x*(4-x)),x=0..4) - Groux Roland, Mar 15 2011.
a(n) = sum of top row terms in M^n, where M is the following infinite square production matrix:
1, 1, 0, 0, 0, 0,...
0, 1, 1, 0, 0, 0,...
1, 1, 1, 1, 0, 0,...
0, 1, 1, 1, 1, 0,...
1, 1, 1, 1, 1, 1....
...
Example: top row of M^3 = (2, 4, 3, 1), sum = 10 = a(3). - Gary W. Adamson, Jul 11, 2011
Conjecture: (n+2)*a(n) -2*(2*n+1)*a(n-1)=0, n>1. - R. J. Mathar, Nov 14 2011
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MAPLE
| 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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PROG
| (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
| Cf. A068875, A068875.
Sequence in context: A192653 A149824 A202135 * A192574 A135336 A149825
Adjacent sequences: A068872 A068873 A068874 * A068876 A068877 A068878
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jun 06 2002
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