A177888 P_n(k) with P_0(z) = z+1 and P_n(z) = z + P_(n-1)(z)*(P_(n-1)(z)-z) for n>1; square array P_n(k), n>=0, k>=0, read by antidiagonals.

1, 2, 1, 3, 3, 1, 4, 5, 7, 1, 5, 7, 17, 43, 1, 6, 9, 31, 257, 1807, 1, 7, 11, 49, 871, 65537, 3263443, 1, 8, 13, 71, 2209, 756031, 4294967297, 10650056950807, 1, 9, 15, 97, 4691, 4870849, 571580604871, 18446744073709551617, 113423713055421844361000443, 1

Offset: 0

Author

Alois P. Heinz, Dec 14 2010

Keywords

• nonn
• tabl

Examples

			Square array P_n(k) begins:
  1,              2,          3,      4,       5,    6,    7,     8, ...
  1,              3,          5,      7,       9,   11,   13,    15, ...
  1,              7,         17,     31,      49,   71,   97,   127, ...
  1,             43,        257,    871,    2209, 4691, 8833, 15247, ...
  1,           1807,      65537, 756031, 4870849,  ...
  1,        3263443, 4294967297,    ...
  1, 10650056950807,        ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..13, flattened
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Crossrefs

Columns k=0-10 give: A000012, A000058(n+1), A000215, A000289(n+1), A000324(n+1), A001543(n+1), A001544(n+1), A067686, A110360(n+1), A110368(n+1), A110383(n+1).

Rows n=0-2 give: A000027(k+1), A005408, A056220(k+1).

Main diagonal gives A252730.

Coefficients of P_n(z) give: A177701.

Programs

Maple

p:= proc(n) option remember;
      z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
seq(seq(p(n)(d-n), n=0..d), d=0..8);

Mathematica

p[n_] := p[n] = Function[z, z + If [n == 0, 1, p[n-1][z]*(p[n-1][z]-z)] ]; Table [Table[p[n][d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)