A103525 Triangle read by rows: T(n,k) is the coefficient of t^k (k>=0) in the polynomial P[n,t] defined by P[1,t] = P[2,t] = 1, P[3,t] = 1+t, P[n,t] = P[n-1,t] + P^2[n-2,1] for n >= 4.

1, 1, 1, 1, 2, 1, 3, 3, 1, 7, 7, 2, 16, 25, 17, 6, 1, 65, 123, 94, 34, 5, 321, 923, 1263, 1076, 626, 254, 70, 12, 1, 4546, 16913, 28612, 28620, 18476, 7876, 2166, 352, 26, 107587, 609479, 1691387, 3050910, 4001833, 4044516, 3255042, 2126032, 1138124, 500806

Offset

1,5

Comments

T(n,k) is the number of certain types of trees (see the Duke et al. reference) of height n and having k branch nodes at level n-1. Row n has 2^(ceiling(n/2)-2)+1 terms (n >= 3). Row sums yield A000278. T(n,0) = A000278(n-1) for n >= 2.

Links

Table of n, a(n) for n=1..50.

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

Formula

T(1,0)=1; T(2,0)=1; T(3,0)=T(3,1)=1; T(n,k)=0 for k >= ceiling(n/2); T(n,k) = T(n-1, k) + Sum_{j=0..k} T(n-2, j)*T(n-2, k-j) for n >= 4.

Example

P[5,t] = 3 + 3*t + t^2; therefore T(3,0)=3, T(3,1)=3, T(3,2)=1.
Triangle begins:
   1;
   1;
   1,  1;
   2,  1;
   3,  3,  1;
   7,  7,  2;
  16, 25, 17,  6,  1;

Maple

P[1]:=1:P[2]:=1:P[3]:=1+t:for n from 4 to 13 do P[n]:=sort(expand(P[n-1]+P[n-2]^2)) od:for n from 1 to 11 do seq(coeff(t*P[n], t^k), k=1..2^(ceil(n/2)-2)+1) od; # yields sequence in triangular form

Crossrefs

Cf. A000278.

Sequence in context: A271238 A186740 A353585 * A294432 A121436 A330694

Adjacent sequences: A103522 A103523 A103524 * A103526 A103527 A103528

Keyword

nonn,tabf

Author

Emeric Deutsch, Mar 21 2005

Status

approved