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A110110
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Number of symmetric Schroeder paths of length 2n (A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis).
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1
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1, 2, 4, 8, 18, 38, 88, 192, 450, 1002, 2364, 5336, 12642, 28814, 68464, 157184, 374274, 864146, 2060980, 4780008, 11414898, 26572086, 63521352, 148321344, 354870594, 830764794, 1989102444, 4666890936, 11180805570, 26283115038
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n)=A026003(n-1)+A026003(n) (n>=1; indeed, every symmetric Schroeder path of length 2n is either a left factor L of length n-1 of a Schroeder path, followed by a H=(2,0) step and followed by the mirror image of L, or it is a left factor of length n of a Schroeder paths, followed by its mirror image).
a(2*n) = A050146(n+1). - Michael Somos Feb 07 2011
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FORMULA
| G.f.: (1 + x) * ( -1 + sqrt( 1 - 6*x^2 + x^4) / (1 - 2*x - x^2)) / (2*x). - Michael Somos Feb 07 2011
G.f.=(1+z)R(z^2)/[1-zR(z^2)], where R=1+zR+zR^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers.
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EXAMPLE
| a(2)=4 because we have HH, UDUD, UHD and UUDD.
1 + 2*x + 4*x^2 + 8*x^3 + 18*x^4 + 38*x^5 + 88*x^6 + 192*x^7 + 450*x^8 + ...
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MAPLE
| RR:=(1-z^2-sqrt(1-6*z^2+z^4))/2/z^2: G:=(1+z)*RR/(1-z*RR): Gser:=series(G, z=0, 36): 1, seq(coeff(Gser, z^n), n=1..33);
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x) * ( -1 + sqrt( 1 - 6*x^2 + x^4 + x^2 * O(x^n)) / (1 - 2*x - x^2)) / (2*x), n))} /* Michael Somos Feb 07 2011 */
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CROSSREFS
| Cf. A026003, A006318.
Sequence in context: A092507 A024415 A018096 * A056362 A086585 A052910
Adjacent sequences: A110107 A110108 A110109 * A110111 A110112 A110113
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 12 2005
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