A177701 Triangle of coefficients of polynomials P_n(z) defined by the recursion P_0(z) = z+1; for n>=1, P_n(z) = z + Product_{k=0..n-1} P_k(z).

1, 1, 2, 1, 2, 4, 1, 4, 14, 16, 8, 1, 16, 112, 324, 508, 474, 268, 88, 16, 1, 256, 3584, 22912, 88832, 233936, 443936, 628064, 675456, 557492, 353740, 171644, 62878, 17000, 3264, 416, 32, 1, 65536, 1835008, 24576000, 209715200, 1281482752, 5974786048, 22114709504, 66752724992

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

Cf. A000058, A000215, A000289, A000324, A001543, A001544, A067686, A110360, A000027, A005408, A056220, A177888. First column gives: A165420.

Sequence in context: A261356 A244419 A157751 * A119765 A303325 A077901

Adjacent sequences: A177698 A177699 A177700 * A177702 A177703 A177704

Keyword

nonn,tabf

Author

Vladimir Shevelev, Dec 11 2010

Extensions

More terms from Alois P. Heinz, Dec 13 2010

Status

approved