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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.

%I A177888 #37 Jan 09 2025 09:55:50
%S A177888 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,
%T A177888 3263443,1,8,13,71,2209,756031,4294967297,10650056950807,1,9,15,97,
%U A177888 4691,4870849,571580604871,18446744073709551617,113423713055421844361000443,1
%N 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.
%H A177888 Alois P. Heinz, <a href="/A177888/b177888.txt">Antidiagonals n = 0..13, flattened</a>
%H A177888 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
%H A177888 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%e A177888 Square array P_n(k) begins:
%e A177888   1,              2,          3,      4,       5,    6,    7,     8, ...
%e A177888   1,              3,          5,      7,       9,   11,   13,    15, ...
%e A177888   1,              7,         17,     31,      49,   71,   97,   127, ...
%e A177888   1,             43,        257,    871,    2209, 4691, 8833, 15247, ...
%e A177888   1,           1807,      65537, 756031, 4870849,  ...
%e A177888   1,        3263443, 4294967297,    ...
%e A177888   1, 10650056950807,        ...
%p A177888 p:= proc(n) option remember;
%p A177888       z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
%p A177888     end:
%p A177888 seq(seq(p(n)(d-n), n=0..d), d=0..8);
%t A177888 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 *)
%Y A177888 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).
%Y A177888 Rows n=0-2 give: A000027(k+1), A005408, A056220(k+1).
%Y A177888 Main diagonal gives A252730.
%Y A177888 Coefficients of P_n(z) give: A177701.
%K A177888 nonn,tabl
%O A177888 0,2
%A A177888 _Alois P. Heinz_, Dec 14 2010