This site is supported by donations to The OEIS Foundation. Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A191314 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n and height k. 6
 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 7, 2, 1, 12, 6, 1, 1, 20, 12, 2, 1, 33, 27, 8, 1, 1, 54, 53, 16, 2, 1, 88, 108, 44, 10, 1, 1, 143, 208, 88, 20, 2, 1, 232, 405, 208, 65, 12, 1, 1, 376, 768, 415, 130, 24, 2, 1, 609, 1459, 908, 350, 90, 14, 1, 1, 986, 2734, 1804, 700, 180, 28, 2, 1, 1596, 5117, 3776, 1700, 544, 119, 16, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row n has 1 + floor(n/2) entries. Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n). T(n,1) = A000071(n+1) (Fibonacci numbers minus 1). Sum_{k>=0} k * T(n,k) = A191315(n). Extracting the even numbered rows, we obtain triangle A205946 with row sums A000984. The odd numbered rows yield triangle A205945 with row sums A001700. - Gary W. Adamson, Feb 01 2012 LINKS Alois P. Heinz, Rows n = 0..200, flattened FORMULA G.f.: The g.f. of column k is z^{2k}/(F[k]*F[k+1]), where F[k] are polynomials in z defined by F=1, F=1-z, F[k]=F[k-1]-z^2*F[k-2] for k>=2. The coefficients of these polynomials form the triangle A108299. Rows may be obtained by taking finite differences of A205573 columns from the top -> down. - Gary W. Adamson, Feb 01 2012 EXAMPLE T(5,2) = 2 because we have HUUDD and UUDDH, where U=(1,1), D=(1,-1), H=(1,0). Triangle starts: 1; 1; 1,  1; 1,  2; 1,  4,  1; 1,  7,  2; 1, 12,  6, 1; 1, 20, 12, 2; 1, 33, 27, 8, 1; MAPLE F := 1: F := 1-z: for k from 2 to 12 do F[k] := sort(expand(F[k-1]-z^2*F[k-2])) end do: for k from 0 to 11 do h[k] := z^(2*k)/(F[k]*F[k+1]) end do: T := proc (n, k) options operator, arrow: coeff(series(h[k], z = 0, 20), z, n) end proc: for n from 0 to 16 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form # second Maple program: b:= proc(x, y) option remember; `if`(y>x or y<0, 0, `if`(x=0, 1,       (p->add(coeff(p, z, i)*z^max(i, y), i=0..degree(p, z)))       (b(x-1, y-1))+ b(x-1, y+1)+`if`(y=0, b(x-1, y), 0)))     end: T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0)): seq(T(n), n=0..14);  # Alois P. Heinz, Mar 12 2014 MATHEMATICA b[x_, y_] := b[x, y] = If[y>x || y<0, 0, If[x==0, 1, Function [{p}, Sum[ Coefficient[p, z, i]*z^Max[i, y], {i, 0, Exponent[p, z]}]][b[x-1, y-1]] + b[x-1, y+1] + If[y==0, b[x-1, y], 0]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *) CROSSREFS Cf. A001405, A000071, A191315. Cf. A205573, A205945, A001700, A205946, A000984. Sequence in context: A211232 A137633 A168533 * A191306 A191525 A066633 Adjacent sequences:  A191311 A191312 A191313 * A191315 A191316 A191317 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, May 31 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)