A191395 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0)-steps at positive heights) for which the sum of the heights of its base pyramid is k. A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis. Here U=(1,1) and D=(1,-1).

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 2, 5, 9, 4, 3, 6, 14, 12, 8, 9, 20, 25, 8, 13, 14, 27, 44, 28, 31, 29, 40, 70, 66, 16, 49, 54, 62, 104, 129, 64, 109, 115, 116, 159, 225, 168, 32, 170, 212, 217, 250, 363, 360, 144, 371, 430, 445, 444, 581, 681, 416, 64, 581, 772, 854, 820, 938, 1182, 968, 320

Offset: 0

Author

Emeric Deutsch, Jun 04 2011

Keywords

• nonn
• tabf

Comments

Row n has 1+ceiling(n/2) entries.

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

T(n,0) = A191393(n).

Sum_{k>=0} k*T(n,k) = A191397(n).

Examples

			T(4,2)=2 because we have UDUD and UUDD, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
  1;
  1;
  1,  1;
  1,  2;
  1,  3,  2;
  1,  4,  5;
  2,  5,  9,  4;
  3,  6, 14, 12;
  8,  9, 20, 25,  8;
		

Crossrefs

Cf. A001405, A191393, A191397.

Programs

Maple

eq := G = 1+z*G+z^2*G*(c+t/(1-t*z^2)-1/(1-z^2)): c := ((1-sqrt(1-4*z^2))*1/2)/z^2: g := simplify(solve(eq, G)): gser := simplify(series(g, z = 0, 19)): for n from 0 to 15 do P[n] := sort(expand(coeff(gser, z, n))) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Formula

G.f.: G=G(t,z) satisfies G = 1+z*G+z^2*G*(c+t/(1-t*z^2)-1/(1-z^2)), where c = (1-sqrt(1-4*z^2))/(2*z^2) (the Catalan function with argument z^2).