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%I A177701 #32 Apr 07 2021 02:49:57
%S A177701 1,1,2,1,2,4,1,4,14,16,8,1,16,112,324,508,474,268,88,16,1,256,3584,
%T A177701 22912,88832,233936,443936,628064,675456,557492,353740,171644,62878,
%U A177701 17000,3264,416,32,1,65536,1835008,24576000,209715200,1281482752,5974786048,22114709504,66752724992
%N 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).
%C A177701 Length of the first row is 2; for i>=2, length of the i-th row is 2^{i-2}+1.
%H A177701 Alois P. Heinz, <a href="/A177701/b177701.txt">Table of n, a(n) for n = 1..1035</a>
%H A177701 A. V. Aho and N. J. A. Sloane, <a href="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, <a href="http://neilsloane.com/doc/doubly.html">alternative link</a>.
%F A177701 Another recursion is: P_n(z)=z+P_(n-1)(z)(P_(n-1)(z)-z).
%F A177701 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)}.
%e A177701 Triangle begins:
%e A177701 1,    1;
%e A177701 2,    1;
%e A177701 2,    4,   1;
%e A177701 4,   14,  16,   8,   1;
%e A177701 16, 112, 324, 508, 474, 268, 88, 16, 1;
%p A177701 p:= proc(n) option remember;
%p A177701        z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
%p A177701     end:
%p A177701 deg:= n-> `if`(n=0, 1, 2^(n-1)):
%p A177701 T:= (n,k)-> coeff(p(n)(z), z, deg(n)-k):
%p A177701 seq(seq(T(n,k), k=0..deg(n)), n=0..6); # _Alois P. Heinz_, Dec 13 2010
%t A177701 P[0][z_] := z + 1;
%t A177701 P[n_][z_] := P[n][z] = z + Product[P[k][z], {k, 0, n-1}];
%t A177701 row[n_] := CoefficientList[P[n][z], z] // Reverse;
%t A177701 Table[row[n], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, Jun 11 2018 *)
%Y A177701 Cf. A000058, A000215, A000289, A000324, A001543, A001544, A067686, A110360, A000027, A005408, A056220, A177888. First column gives: A165420.
%K A177701 nonn,tabf
%O A177701 1,3
%A A177701 _Vladimir Shevelev_, Dec 11 2010
%E A177701 More terms from _Alois P. Heinz_, Dec 13 2010

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