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Search: id:A114713
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| A114713 |
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Number of ascents in all peakless Motzkin paths of length n+3. A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis. An ascent in a Motzkin path is a maximal sequence of consecutive U steps. |
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+0
2
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| 1, 3, 7, 18, 46, 116, 294, 746, 1894, 4816, 12262, 31258, 79777, 203833, 521337, 1334690, 3420039, 8770891, 22510949, 57817420, 148599626, 382165858, 983430962, 2532082308, 6522876601, 16811813391, 43350264107, 111830286218
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OFFSET
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0,2
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COMMENT
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a(n)=Sum(k*A114712(n+3,k),k=0..1+floor(n/3)).
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REFERENCES
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I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes d'enumeration en biologie moleculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
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FORMULA
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G.f.=[1-2z+z^2-2z^3+z^4-(1-z+z^2)sqrt(1-2z-z^2-2z^3+z^4)]/[2z^4*sqrt(1-2z-z^2-2z^3+z^4)].
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EXAMPLE
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a(2)=7 because in the 8 (=A004148(5)) peakless Motzkin paths of length 5, namely HHHHH, (U)HDHH, (U)HHDH, (U)HHHD, H(U)HDH, H(U)HHD, HH(U)HD and (UU)HDD, we have altogether 7 ascents (shown between parentheses).
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MAPLE
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G:=(1-2*z+z^2-2*z^3+z^4-(1-z+z^2)*sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^4/(1-2*z-z^2-2*\ z^3+z^4)^(1/2): Gser:=series(G, z=0, 40): 1, seq(coeff(Gser, z^n), n=1..32);
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CROSSREFS
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Cf. A004148, A114712.
Sequence in context: A027967 A000226 A036883 this_sequence A116413 A078058 A052960
Adjacent sequences: A114710 A114711 A114712 this_sequence A114714 A114715 A114716
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2005
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