A276436 Generating function f(x)=(x+(x+(x+(x+(x+...)^5)^4)^3)^2)^1 is the limit as n->infinity of (f_1(x)=x, f_2(x)=x+x^2, f_3(x)=x+(x+x^3)^2, f_4(x)=x+(x+(x+x^4)^3)^2, ...).

0, 1, 1, 0, 2, 0, 1, 6, 0, 6, 6, 24, 15, 26, 48, 56, 240, 60, 303, 504, 780, 1002, 1776, 3246, 3601, 7826, 7500, 18980, 26874, 38130, 56196, 99360, 153636, 210084, 390348, 486420, 900428, 1310118, 2064612, 3073008, 4825138, 7558008, 11428162, 18596886, 26006031, 43625940, 65162736, 100027728, 152897710, 242895050

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

Sequence in context: A331787 A321686 A055925 * A161121 A021500 A175197

Adjacent sequences: A276433 A276434 A276435 * A276437 A276438 A276439

Keyword

nonn

Author

Yuriy Sibirmovsky, Sep 02 2016

Status

approved