|
| |
|
|
A307397
|
|
G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k*A(x)^k/(1 + x^k*A(x)^k).
|
|
4
|
|
|
|
1, 1, 1, 3, 8, 18, 50, 150, 429, 1258, 3835, 11740, 36148, 112856, 355318, 1124582, 3582186, 11477162, 36939043, 119387415, 387393424, 1261422550, 4120343870, 13498085604, 44337516318, 145993301239, 481812344551, 1593439356575, 5280074015618, 17528034861180
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} A048272(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} A048272(k)*x^k)).
a(n) ~ c * d^n / n^(3/2), where d = 3.494393309755265712092948136162079492013891957890570364999827377394989... and c = 0.4966863488644340281558816065879601408815044380350316600850227488... - Vaclav Kotesovec, Sep 27 2023
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 8*x^4 + 18 x^5 + 50*x^6 + 150*x^7 + 429*x^8 + 1258*x^9 + 3835*x^10 + ...
|
|
|
MATHEMATICA
|
terms = 30; A[_] = 0; Do[A[x_] = 1 + Sum[x^k A[x]^k/ (1 + x^k A[x]^k), {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 30; A[_] = 0; Do[A[x_] = 1 + Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 30; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
(* Calculation of constant d: *) val = r /. FindRoot[{Log[1 - r*s] + QPolyGamma[0, 1 - I*Pi/Log[r*s], r*s] == (1 - s)*Log[r*s], (1 - s)/s - I*Pi*QPolyGamma[1, 1 - I*Pi/Log[r*s], r*s] / (s*Log[r*s]^2) + r*(1/(1 - r*s) - Derivative[0, 0, 1][QPolyGamma][0, 1 - I*Pi/Log[r*s], r*s]) == Log[r*s]}, {r, 1/3}, {s, 2}, WorkingPrecision -> 100]; N[1/Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* Vaclav Kotesovec, Sep 27 2023 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
