OFFSET
1,8
COMMENTS
lim(N->infinity)p_N(x)/prod(n = 0, N-1)p_n(x) = f(x), with f(x) from A109087.
LINKS
Robert Israel, Table of n, a(n) for n = 1..9957
FORMULA
p_0(x) = x, p_(n+1)(x) = p_n(x)^2 + prod(i = 0, n-1)p_i(x)^(n-i), n >= 0. p_n(x) = sum(i = 0, 2^n)t(n, i)*x^i, n >= 0. a(2^(n+1)+n-i)=t(n, i), n>=0, 0<=i<=2^n.
EXAMPLE
p_0(x) = x, p_1(x) = x^2 + 1, p_2(x) = x^4 + 2*x^2 + x + 1,
p_3(x) = x^8 + 4*x^6 + 2*x^5 + 7*x^4 + 4*x^3 + 6*x^2 + 2*x + 1
p_4(x) = x^16 + 8*x^14 + 4*x^13 + 30*x^12 + 25*x^11 + 72*x^10 + 68*x^9 + 116*x^8 + 102*x^7 + 118*x^6 + 84*x^5 + 67*x^4 + 33*x^3 + 16*x^2 + 4*x + 1
MAPLE
p[0]:= x:
for n from 1 to 7 do
p[n]:= expand(p[n-1]^2 + mul(p[i]^(n-1-i), i=0..n-2))
od:
for n from 0 to 7 do
seq(coeff(p[n], x, j), j=degree(p[n]) .. 0, -1)
od; # Robert Israel, Jun 23 2020
PROG
(PARI) N=5; p=x; q=1; r=1; for(n=0, N, print(Vec(p)); q*=r; r*=p; p=p^2+q)
CROSSREFS
KEYWORD
AUTHOR
Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jun 19 2005
STATUS
approved