A097854 Triangle read by rows: T(n,k) = number of Motzkin paths of length n and having abscissa of first return (i.e., first down step hitting the x-axis) equal to k (k>0); T(n,0)=1 (accounts for the paths consisting only of level steps).

1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 2, 4, 1, 0, 4, 4, 4, 8, 1, 0, 9, 8, 8, 8, 17, 1, 0, 21, 18, 16, 16, 17, 38, 1, 0, 51, 42, 36, 32, 34, 38, 89, 1, 0, 127, 102, 84, 72, 68, 76, 89, 216, 1, 0, 323, 254, 204, 168, 153, 152, 178, 216, 539, 1, 0, 835, 646, 508, 408, 357, 342, 356, 432, 539, 1374

Offset

0,10

Comments

Row sums are the Motzkin numbers (A001006).

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

G.f.: (1-t*z+t^2*z^2*M(t*z)*M(z) - t^2*z^3*M(t*z)*M(z))/(1-z-t*z+t*z^2), where M(z)=(1-z-sqrt(1-2*z-3*z^2))/(2*z^2) is the g.f. of the Motzkin numbers.

T(n,k) = m(n-k)*Sum_{j=0..k-2} m(j), where m(n) = A001006(n) are the Motzkin numbers.

Example

Triangle starts:
  1;
  1, 0;
  1, 0, 1;
  1, 0, 1, 2;
  1, 0, 2, 2, 4;
  1, 0, 4, 4, 4, 8;
Row n has n+1 terms.
T(5,3)=4 because the Motzkin paths of length 5 and having abscissa of first return equal to 3 are HU(D)HH, HU(D)UD, UH(D)HH and UH(D)UD (first returns to axis shown between parentheses); here U=(1,1), H=(1,0) and D=(1,-1).

Maple

G:=(1-t*z+t^2*z^2*M(t*z)*M(z)-t^2*z^3*M(t*z)*M(z))/(1-z-t*z+t*z^2): M:=z->(1-z-sqrt(1-2*z-3*z^2))/2/z^2: Gser:=simplify(series(G, z=0, 14)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser, z^n) od: seq(seq(coeff(t*P[n], t^k), k=1..n+1), n=0..12); M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: Mser:=series(M, z=0, 15): m[0]:=1: for n from 1 to 12 do m[n]:=coeff(Mser, z^n) od: T:=proc(n, k) if k=0 then 1 elif k<=n then m[n-k]*sum(m[j], j=0..k-2) else 0 fi end: TT:=(n, k)->T(n-1, k-1): matrix(11, 11, TT); # generates the triangle:

Mathematica

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; t[n_, 0] = 1; t[n_, k_] := m[n - k]*Sum[m[j], {j, 0, k - 2}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 10 2013 *)

Crossrefs

Cf. A001006.

Sequence in context: A316658 A189962 A308321 * A161515 A145580 A144219

Adjacent sequences: A097851 A097852 A097853 * A097855 A097856 A097857

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 31 2004

Extensions

Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013

Terms a(75) and beyond from G. C. Greubel, Oct 23 2017

Status

approved