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A098057
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Number of peakless Motzkin paths with no U H^j U, no D H^j D and no D H^jU (j>0), where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).
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2
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1, 1, 1, 2, 4, 8, 15, 27, 48, 84, 147, 257, 451, 796, 1413, 2526, 4544, 8226, 14978, 27417, 50434, 93183, 172865, 321857, 601263, 1126644, 2116968, 3987960, 7530200, 14249649, 27019301, 51327965, 97676156, 186177568, 355406479, 679425009
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: [1-z+z^2-4z^3+2z^4-sqrt(1-2z-z^2+2z^3+z^4-4z^5+4z^6)]/[2z^2*(1-z)^3].
D-finite with recurrence (n+2)*a(n) +(-5*n-7)*a(n-1) +2*(4*n+5)*a(n-2) +(-2*n-13)*a(n-3) +3*(-2*n+5)*a(n-4) +18*(1)*a(n-5) +(17*n-101)*a(n-6) +(-25*n+154)*a(n-7) +2*(8*n-53)*a(n-8) +4*(-n+7)*a(n-9)=0. - R. J. Mathar, Jul 26 2022
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EXAMPLE
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a(4)=4 because we have HHHH, UHDU, HUHD and UHHD; a(6)=15 because from all 17 peakless Motzkin paths of length 6 (see A004148) only (UHU)HDD and UUH(DHD) do not qualify.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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