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A080937
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Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.
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15
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1, 1, 2, 5, 14, 42, 131, 417, 1341, 4334, 14041, 45542, 147798, 479779, 1557649, 5057369, 16420730, 53317085, 173118414, 562110290, 1825158051, 5926246929, 19242396629, 62479659622, 202870165265, 658715265222, 2138834994142
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| With interpolated zeros (1,0,1,0,2,...), counts closed walks of length n at start or end node of P_6. The sequence (0,1,0,2,...) counts walks of length n between the start and second node. - Paul Barry (pbarry(AT)wit.ie), Jan 26 2005
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REFERENCES
| P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, Arxiv preprint arXiv:1109.3641, 2011
Aleksandar Ilic and Andreja Ilic, On the number of restricted Dyck paths, Filomat 25:3 (2011), 191-201; DOI: 10.2298/FIL1103191I, http://operator.pmf.ni.ac.rs/www/pmf/publikacije/filomat/2011/F25-3-2011/F25-3-17.pdf
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (5,-6,1).
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FORMULA
| a(n) =A080934(n, 5)
G.f.: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3). - Ralf Stephan (ralf(AT)ark.in-berlin.de), May 13 2003
a(n) = 5*a(n-1)-6*a(n-2)+a(n-3) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 11 2004
a(n)=A096976(2*n) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Nov 02 2005
a(n)=(4/7-4/7*cos(1/7*Pi)^2)*(4*(cos(Pi/7))^2)^n+(1/7-2/7*cos(1/7*Pi)+4/7*cos(1/7*Pi)^2)*(4*(cos(2*Pi/7))^2)^n+(2/7+2/7*cos(1/7*Pi))*(4*(cos(3*Pi/7))^2)^n for n>=0. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Apr 19 2010]
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PROG
| (PARI) a=vector(99); a[1]=1; a[2]=2; a[3]=5; for(n=4, #a, a[n]=5*a[n-1]-6*a[n-2]+a[n-3]); a \\ Charles R Greathouse IV, Jun 10 2011
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CROSSREFS
| Cf. A000007, A000012, A011782, A001519, A007051, A080937, A024175, A080938, A033191 which essentially provide the same sequence for different limits and tend to A000108.
Cf. A094790, A094789, A005021.
Sequence in context: A148327 A092493 A148328 * A196417 A054392 A006930
Adjacent sequences: A080934 A080935 A080936 * A080938 A080939 A080940
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KEYWORD
| nonn,easy
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Feb 25 2003
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