login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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.
4
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
0,3
COMMENTS
Row n has 1+n(n+1)/2 terms.
Row sums are the Catalan numbers (A000108).
Column sums yield A095830.
Sum(k*T(n,k),k>=0)=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).
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
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
FORMULA
G.f.=G(t,z) satisfies G(t,z)=1 + 2tzG(t,tz) + [tzG(t,tz)]^2. The row generating polynomials P[n]=P[n](t) (n=1,2,...) are given by P[n]=t^n (2P[n-1] + sum(P[i]P[n-2-i], i=0..n-2); P[0]=1.
EXAMPLE
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;
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. *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Mar 20 2008
STATUS
approved