OFFSET
1,3
COMMENTS
Length of the first row is 2; for i>=2, length of the i-th row is 2^{i-2}+1.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1035
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437, alternative link.
FORMULA
Another recursion is: P_n(z)=z+P_(n-1)(z)(P_(n-1)(z)-z).
Private values: P_n(0)=1; P_n(-1)=delta_(n,0)-1; {P_n(1)}=A000058; {P_n(2)}=A000215; {P_n(3)}={A000289(n+1)}; {P_n(4)}={A000324(n+1)}; {P_n(5)}={A001543(n+1)}; {P_n(6)}={A001544(n+1)}; {P_n(7)}={A067686(n)}; {P_n(8)}={A110360(n)}; {P_0(n)}={A000027(n+1)}; {P_1(n)}={A005408(n)}; {P_2(n)}={A056220(n+1)}.
EXAMPLE
Triangle begins:
1, 1;
2, 1;
2, 4, 1;
4, 14, 16, 8, 1;
16, 112, 324, 508, 474, 268, 88, 16, 1;
MAPLE
p:= proc(n) option remember;
z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
end:
deg:= n-> `if`(n=0, 1, 2^(n-1)):
T:= (n, k)-> coeff(p(n)(z), z, deg(n)-k):
seq(seq(T(n, k), k=0..deg(n)), n=0..6); # Alois P. Heinz, Dec 13 2010
MATHEMATICA
P[0][z_] := z + 1;
P[n_][z_] := P[n][z] = z + Product[P[k][z], {k, 0, n-1}];
row[n_] := CoefficientList[P[n][z], z] // Reverse;
Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Vladimir Shevelev, Dec 11 2010
EXTENSIONS
More terms from Alois P. Heinz, Dec 13 2010
STATUS
approved