A103522 Triangle read by rows: T(n,k) is the coefficient of t^k (k>=1) in the polynomial P[n,t] defined by P[1,t]=P[2,t]=t^2, P[n,t]=tP[n-1,t]+t^2*P^2[n-2,t] (n>=3).

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 4, 5, 5, 5, 7, 8, 7, 6, 6, 4, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 4, 6, 7, 8, 11, 17, 21, 25, 28, 31, 31, 33, 32, 27, 18, 10, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset

1,22

Comments

T(n,k) is the number of certain types of trees (see the Duke et al. reference) of height n and having k edges. Row n contains A027383(n-1) terms, the first n-1 of which are 0. Row sums yield A000278.

Links

Table of n, a(n) for n=1..99.

W. Duke, Stephen J. Greenfield and Eugene R. Speer, Properties of a Quadratic Fibonacci Recurrence, J. Integer Sequences, 1998, #98.1.8.

Example

P[3,t]=t^3+t^4; therefore T(3,1)=0, T(3,2)=0, T(3,3)=1, T(3,4)=1.
Triangle begins:
1;
0,1;
0,0,1,1;
0,0,0,1,1,1;
0,0,0,0,1,1,1,1,2,1;
0,0,0,0,0,1,1,1,1,3,3,3,2,1;

Maple

P[1]:=t:P[2]:=t^2:for n from 3 to 12 do P[n]:=sort(expand(t*P[n-1]+t^2*P[n-2]^2)) od: d[1]:=1: d[2]:=2: for n from 3 to 20 do d[n]:=2*d[n-2]+2 od: for n from 1 to 9 do seq(coeff(P[n], t^k), k=1..d[n]) od; # yields sequence in triangular form

Crossrefs

Cf. A000278, A027383.

Sequence in context: A015010 A016011 A263338 * A101108 A279280 A284093

Adjacent sequences: A103519 A103520 A103521 * A103523 A103524 A103525

Keyword

nonn,tabf

Author

Emeric Deutsch, Mar 21 2005

Status

approved