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 A190170 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k UHD's starting at level 0; here U=(1,1), H=(1,0), and D=(1,-1). 2
 1, 1, 1, 1, 1, 2, 2, 5, 3, 12, 4, 1, 27, 7, 3, 60, 16, 6, 135, 39, 10, 1, 309, 92, 18, 4, 717, 212, 39, 10, 1680, 488, 94, 20, 1, 3966, 1135, 228, 39, 5, 9423, 2670, 543, 84, 15, 22518, 6336, 1282, 200, 35, 1, 54091, 15132, 3036, 492, 75, 6, 130540, 36327, 7245, 1203, 166, 21 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row n contains 1+floor(n/3) entries. Sum of entries in row n = A004148(n). T(n,0)=A190171(n). Sum(k*T(n,k),k>=0)=A089735(n-3). LINKS FORMULA G.f. G=G(t,z) is obtained by elimitaing S from the equations G=1+zG+z^2*G(S-1-z+tz) and S=1+zS+z^2*S(S-1). EXAMPLE T(6,2)=1 because we have UHDUHD. Triangle starts: 1; 1; 1; 1,1; 2,2; 5,3; 12,4,1; 27,7,3; MAPLE p1 := G-1-z*G-z^2*G*(S-1-z+t*z): p2 := S-1-z*S-z^2*S*(S-1): r := resultant(p1, p2, S): g := RootOf(r, G): Gser := simplify(series(g, z = 0, 21)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form CROSSREFS Cf. A004148, A190171, A089735 Sequence in context: A308096 A006800 A241820 * A147524 A113177 A344507 Adjacent sequences:  A190167 A190168 A190169 * A190171 A190172 A190173 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, May 06 2011 STATUS approved

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Last modified September 25 05:14 EDT 2021. Contains 347652 sequences. (Running on oeis4.)