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A045868
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G.f.: ((1-x-sqrt(1-6x+5x^2))/(2x))^2.
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4
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1, 2, 7, 26, 101, 406, 1676, 7066, 30302, 131782, 579867, 2576982, 11550237, 52152330, 237005385, 1083211410, 4975796735, 22960105510, 106377393365, 494674698190, 2308015808015, 10801388134690, 50691017885290
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Convolution of A002212 with itself.
Number of skew Dyck paths of semilength n+1 starting with UU. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=7 because we have UUDDUD, UUDUDD, UUDUDL, UUUDDD, UUUDDL, UUUDLD and UUUDLL. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 11 2007
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REFERENCES
| S. J. Cyvin et al., Enumeration and classification of certain polygonal systems...: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
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LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
| a(n)=(2/n)*sum(binomial(n, j)*binomial(2j+1, j-1), j=1..n) for n>=1.
(n+2)*a(n) = (6*n+2)*a(n-1)-(5*n-10)*a(n-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 16 2004
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MAPLE
| a := n->(2/n)*sum(binomial(n, j)*binomial(2*j+1, j-1), j=1..n): 1, seq(a(n), n=1..22);
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PROG
| (PARI) a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))^2/4, n+2)
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CROSSREFS
| T(n, n-1) where T is A055450.
Essentially the first differences of A002212 and A025238.
Sequence in context: A049775 A101850 A176280 * A171711 A129482 A150536
Adjacent sequences: A045865 A045866 A045867 * A045869 A045870 A045871
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 11 2007
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