A103524 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]=t, P[2,t]=t^2, P[n,t]=tP[n-1,t]+t^2*P^2[n-2,1].

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 4, 1, 1, 1, 0, 9, 4, 1, 1, 1, 0, 49, 9, 4, 1, 1, 1, 0, 256, 49, 9, 4, 1, 1, 1, 0, 4225, 256, 49, 9, 4, 1, 1, 1, 0, 103041, 4225, 256, 49, 9, 4, 1, 1, 1, 0, 20666116, 103041, 4225, 256, 49, 9, 4, 1, 1, 1, 0, 11574962569, 20666116, 103041

Offset

1,12

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 from the root to the first branch node (k edges if there are no branch nodes). Row sums yield A000278.

Links

Table of n, a(n) for n=1..70.

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

Formula

T(n, k)=0 for k>n; T(n, n)=1; T(n, 1)=0 for n>=2; T(n, k)=A000278(n-k)^2 for 2<=k<=n-1.

Example

P[3,t]=t^2+t^3; therefore T(3,1)=0, T(3,2)=1, T(3,3)=1.
Triangle begins:
1;
0,1;
0,1,1;
0,1,1,1;
0,4,1,1,1;
0,9,4,1,1,1;
0,49,9,4,1,1,1;

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

Crossrefs

Cf. A000278.

Sequence in context: A324564 A276974 A122777 * A110916 A185058 A368474

Adjacent sequences: A103521 A103522 A103523 * A103525 A103526 A103527

Keyword

nonn,tabl

Author

Emeric Deutsch, Mar 21 2005

Status

approved