OFFSET
0,5
COMMENTS
The approximation f_n(x) gives n(n-1)/2 correct terms of the sequence.
Conjectures: the sequence contains no zeros except for a(0), a(3), a(5) and a(8); the sequence is asymptotically monotonic, the ratios of successive terms a(n+1)/a(n) converge to a constant c, 1<c<1.4.
a(4949)/a(4948) = 1.4132183695,
a(11174)/a(11173) = 1.3977587640.
LINKS
Yuriy Sibirmovsky, Table of n, a(n) for n = 0..6000
Yuriy Sibirmovsky, A sequence of coefficients of x+(x+(x+(x+(x+(x+…)^6)^5)^4)^3)^2, Mathematics Stackexchange, Sep 03 2016.
Yuriy Sibirmovsky, Plot of a(n+1)/a(n) for n=0..2414
FORMULA
Another way to represent the generating function:
F(1)(x,y) = y;
F(n)(x,y) = F(n-1)(x,x+y^n), n>1;
f(x) = Lim(n->infinity) F(n)(x,y);
Then: f(x)=Sum_{n>=0} a(n)*x^n. (Due to Markus Scheuer at Mathematics Stackexchange)
EXAMPLE
G.f. = x + x^2 + 2*x^4 + x^6 + 6*x^7 + 6*x^9 + 6*x^10 + 24*x^11 + 15*x^12 + ...
MATHEMATICA
Nm = 20;
F = Table[1, {j, 1, Nm}];
F[[1]] = y;
Do[F[[j + 1]] = F[[j]] /. y -> x + y^(j + 1), {j, 1, Nm - 1}];
f = F[[Nm]] /. y -> 0;
CoefficientList[Series[f, {x, 0, Nm (Nm - 1)/2}], x]
a[ n_] := If[ n < 0, 0, Module[{m, A}, m = Quotient[1 + Sqrt[8 n + 1], 2]; A = x^m + x O[x]^n; Do[ A = (x + A)^(m - i), {i, m - 1}]; SeriesCoefficient[ A, {x, 0, n}]]]; (* Michael Somos, Aug 15 2017 *)
PROG
(PARI) {a(n) = my(A, m); if( n<0, 0, m = (sqrtint(8*n+1) + 1) \ 2; A = x^m + x * O(x^n); for(i=1, m-1, A = (x + A)^(m-i)); polcoeff(A, n))}; /* Michael Somos, Aug 15 2017 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Yuriy Sibirmovsky, Sep 02 2016
STATUS
approved