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A104547
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Number of Schroeder paths of length 2n having no UHD, UHHD, UHHHD, ..., where U=(1,1),D=(1,-1), H=(2,0).
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2
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1, 2, 5, 16, 60, 245, 1051, 4660, 21174, 98072, 461330, 2197997, 10585173, 51443379, 251982793, 1242734592, 6165798680, 30754144182, 154123971932, 775669589436, 3918703613376, 19866054609754, 101029857327802, 515275408644773
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318).
a(n)=A104546(n,0)
Equals binomial transform of A119370. - Paul D. Hanna (pauldhanna(AT)juno.com), May 17 2006
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FORMULA
| G.f.=G=G(z) satisfies G=1+zG+zG[G-z/(1-z)].
G.f.: A(x) = (1-2*x+2*x^2 - sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)). - Paul D. Hanna (pauldhanna(AT)juno.com), May 17 2006
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EXAMPLE
| a(2)=5 because we have HH, HUD, UDH, UDUD and UUDD (UHD does not qualify).
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PROG
| (PARI) {a(n)=polcoeff(2*(1-x)/(1-2*x+2*x^2 + sqrt(1-8*x+16*x^2-12*x^3+4*x^4+x*O(x^n))), n)} - Paul D. Hanna (pauldhanna(AT)juno.com), May 17 2006
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CROSSREFS
| Cf. A006318, A104546.
Cf. A119370.
Sequence in context: A007747 A107283 A059237 * A186999 A000764 A205486
Adjacent sequences: A104544 A104545 A104546 * A104548 A104549 A104550
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2005
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