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 A114580 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k ascents (0<=k<=floor(n/2)); an ascent is a maximal string of upsteps. 1
 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 14, 6, 1, 26, 23, 1, 1, 46, 70, 10, 1, 79, 186, 56, 1, 1, 133, 451, 235, 15, 1, 221, 1025, 825, 115, 1, 1, 364, 2220, 2562, 630, 21, 1, 596, 4634, 7274, 2794, 211, 1, 1, 972, 9396, 19286, 10696, 1456, 28, 1, 1581, 18612, 48450, 36715 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row n contains 1+floor(n/2) terms. Row sums are the Motzkin numbers (A001006). Sum(k*T(n,k),k=0..floor(n/2))=A005774(n-1). LINKS Alois P. Heinz, Rows n = 0..200, flattened Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2. FORMULA G.f.: G(t,z) satisfies G = 1+zG+z^2[t(1+zG)+G-1-zG]G. EXAMPLE T(4,1) = 7 because we have HH(U)D, H(U)DH, H(U)HD, (U)DHH, (U)HDH, (U)HHD and (UU)HH, where U=(1,1), H=(1,0), D=(1,-1) (the ascents are shown between parentheses). Triangle begins: 1; 1; 1,  1; 1,  3; 1,  7,  1; 1, 14,  6; 1, 26, 23, 1; MAPLE G:=1/2*(1-z+z^2-t*z^2-sqrt(1-z^2-2*z+2*z^3-2*z^3*t-2*z^2*t+z^4-2*z^4*t+z^4*t^2))/z^2/(z*t+1-z): Gserz:=simplify(series(G, z=0, 18)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(Gserz, z^n)) od: for n from 0 to 15 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form # second Maple program: b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,       `if`(x=0, 1, expand(`if`(t, 1, z)*b(x-1, y-1, true)       +b(x-1, y+1, false)+b(x-1, y, false))))     end: T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, false)): seq(T(n), n=0..20);  # Alois P. Heinz, Mar 11 2014 MATHEMATICA b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[If[t, 1, z]*b[x-1, y-1, True] + b[x-1, y+1, False] + b[x-1, y, False]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, False]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *) CROSSREFS Cf. A001006, A005774. Sequence in context: A026499 A242114 A143470 * A257597 A097229 A097862 Adjacent sequences:  A114577 A114578 A114579 * A114581 A114582 A114583 KEYWORD nonn,tabf,changed AUTHOR Emeric Deutsch, Dec 09 2005 STATUS approved

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Last modified February 21 07:18 EST 2018. Contains 299390 sequences. (Running on oeis4.)