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Conjecture D-finite with recurrence -4*(n+1)*(n-7)*a(n) +(13*n^2-85*n+28)*a(n-1) +(-7*n^2+52*n-41)*a(n-2) +(5*n^2-41*n+67)*a(n-3) +(-13*n^2+103*n-197)*a(n-4) +(7*n-29)*(n-5)*a(n-5) -(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jul 22 2022
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editing
_Emeric Deutsch (deutsch(AT)duke.poly.edu), _, May 05 2011
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allocated for Emeric DeutschNumber of subwords of type dh^ju (j>=1), where u=(1,1), h=(1,0), and d=(1,-1), in all peakless Motzkin paths of length n (can be easily expressed using RNA secondary structure terminology).
0, 0, 0, 0, 0, 0, 0, 1, 5, 18, 58, 174, 500, 1399, 3843, 10421, 27997, 74699, 198267, 524135, 1381261, 3631068, 9526568, 24954538, 65283648, 170610003, 445484163, 1162396269, 3031267533, 7901082379, 20586262763, 53620039074, 139624131310, 363495081689, 946147596489, 2462387385085
0,9
a(n)=Sum(k*A098083(n,k), k>=0).
G.f.: G(z)=z^5*g^2*(g-1)^2/[(1-z)(1-z^2*g^2)], where g=1+zg+z^2*g(g-1).
a(7)=1 because among the 37 (=A004148(7)) peakless Motzkin paths of length 7 only uh(dhu)hd has a subword of the prescribed type (shown between parentheses).
eq := g = 1+z*g+z^2*g*(g-1): g := RootOf(eq, g): G := z^5*g^2*(g-1)^2/((1-z)*(1-z^2*g^2)): Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 0 .. 35);
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nonn
Emeric Deutsch (deutsch(AT)duke.poly.edu), May 05 2011
approved
proposed
allocated for Emeric Deutsch
allocated
approved