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.
Original entry on oeis.org
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
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, ...
- 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!)
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).
Coefficients of P_n(z) give:
A177701.
-
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);
-
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 *)
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).
Original entry on oeis.org
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
Triangle begins:
1, 1;
2, 1;
2, 4, 1;
4, 14, 16, 8, 1;
16, 112, 324, 508, 474, 268, 88, 16, 1;
...
- 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.
- 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!)
Cf.
A000058,
A000215,
A000289,
A000324,
A001543,
A001544,
A067686,
A110360,
A000027,
A005408,
A056220,
A177888.
-
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
-
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 *)
Showing 1-2 of 2 results.
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