A138157 Triangle read by rows: T(n,k) is the number of binary trees with n edges and path length k; 0<=k<=n*(n+1)/2.

1, 0, 2, 0, 0, 1, 4, 0, 0, 0, 0, 4, 2, 8, 0, 0, 0, 0, 0, 0, 6, 8, 8, 4, 16, 0, 0, 0, 0, 0, 0, 0, 0, 4, 24, 4, 28, 16, 16, 8, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 24, 36, 48, 24, 56, 40, 56, 32, 32, 16, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 60, 72, 144, 26, 168, 104, 128, 64, 176, 80, 112

Offset: 0

Author

Emeric Deutsch, Mar 20 2008

Keywords

• nonn
• tabf

Comments

Row n has 1+n*(n+1)/2 terms.

Row sums are the Catalan numbers (A000108).

Column sums yield A095830.

Sum_{k>=0} k*T(n,k) = A138156(n).

The g.f. B(w,z) in the Knuth reference is related to the above G(t,z) through B(t,z)=1+zG(t,z).

Examples

			T(2,3) = 4 because we have the path trees LL, LR, RL and RR, where L (R) denotes a left (right) edge; each of these four trees has path length 3.
Triangle starts:
  1;
  0,2;
  0,0,1,4;
  0,0,0,0,4,2,8;
  0,0,0,0,0,0,6,8,8,4,16;
  0,0,0,0,0,0,0,0,4,24,4,28,16,16,8,32;
		

References

• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1997, Vol. 1, p. 405 (exercise 5) and p. 595 (solution).

Links

• Alois P. Heinz, Rows n = 0..50, flattened
• Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.

Crossrefs

Cf. A000108, A095830, A138156.

Programs

Maple

P[0]:=1: for n to 7 do P[n]:=sort(expand(t^n*(2*P[n-1]+add(P[i]*P[n-2-i],i= 0..n-2)))) end do: for n from 0 to 7 do seq(coeff(P[n],t,j),j=0..(1/2)*n*(n+1)) end do; # yields sequence in triangular form

Mathematica

p[n_] := p[n] = t^n*(2p[n-1] + Sum[p[i]*p[n-2-i], {i, 0, n-2}]); p[0] = 1; Flatten[ Table[ CoefficientList[ p[n], t], {n, 0, 7}]] (* Jean-François Alcover, Jul 22 2011, after Maple prog. *)

Formula

G.f.: G(t,z) satisfies G(t,z) = 1 + 2*t*z*G(t,t*z) + (t*z*G(t,t*z))^2. The row generating polynomials P[n] = P[n](t) (n=1,2,...) are given by P[n] = t^n*(2*P[n-1] + Sum_{i=0..n-2} P[i]*P[n-2-i]); P[0] = 1.