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A073157
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Left-most column of triangle A073154.
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9
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1, 2, 5, 18, 70, 293, 1283, 5808, 26960, 127628, 613814, 2990681, 14730713, 73229291, 366936231, 1851352820, 9397497758, 47957377934, 245903408244, 1266266092112, 6545667052320, 33954266444498, 176689391245146
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of Schroeder n-paths containing no FFs. A Schroeder n-path (A006318) consists of steps U=(1,1),F=(2,0),D=(1,-1) starting at (0,0), ending at (2n,0), and never going below the x-axis. Example: a(2)=5 counts UFD, UUDD, UDF, FUD, UDUD. [David Callan, Aug 23 2011]
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FORMULA
| Convolution yields sequence A073155 minus the 0-th term: A073155(n+1)=sum{k=0..n} a(k)* a(n-k).
G.f.: A(x) = (1 - sqrt(1 - 4*x*(1+x)^2))/(2*x*(1+x)) satisfies A(x) = (1+x)*(1 + x*A(x)^2); G.f.: A(x) = (1+x)*C(x*(1+x)^2) where C(x) is the Catalan g.f. of A000108. - Paul D. Hanna, Mar 03 2008
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EXAMPLE
| G.f.: A(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 +...
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PROG
| (PARI) {a(n)=local(A=1); for(i=0, n-1, A=(1+x)*(1+x*(A+x*O(x^n))^2)); polcoeff(A, n)} /* Paul D. Hanna, Mar 03 2008 */
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CROSSREFS
| Cf. A073154, A073155, A073156, A073153.
Cf. A000108.
Sequence in context: A150025 A118814 A014271 * A141494 A189843 A045612
Adjacent sequences: A073154 A073155 A073156 * A073158 A073159 A073160
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KEYWORD
| easy,nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jul 29 2002
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EXTENSIONS
| More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Mar 03 2008
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