A386721 Expansion of e.g.f. exp(x/(1 - 9*x^3)^(1/3)).

1, 1, 1, 1, 73, 361, 1081, 93241, 912241, 4907953, 476295121, 7244922961, 58360393081, 6211842488281, 130899060524233, 1435239754046281, 164948740478252641, 4498516738183799521, 63300797606830713121, 7772118657831401082913, 262261735708117281036841

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Vaclav Kotesovec, Graph - the asymptotic ratio (20000 terms)

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} 9^k * binomial(n/3-1,k)/(n-3*k)!.

a(n) == 1 mod 72.

From Vaclav Kotesovec, Sep 03 2025: (Start)

a(n) = (36*n^3 - 324*n^2 + 1008*n - 1079)*a(n-3) - 162*(n-6)*(n-5)*(n-4)*(n-3)*(3*n^2 - 27*n + 64)*a(n-6) + 2916*(n-9)*(n-8)*(n-7)*(n-6)^3*(n-5)*(n-4)*(n-3)*a(n-9) - 6561*(n-12)*(n-11)*(n-10)*(n-9)^2*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*a(n-12).

a(n) ~ 3^(2*n/3 - 1/4) * exp(4*3^(-3/2)*n^(1/4) - n) * n^(n - 3/8) / 2. (End)

Mathematica

nmax = 20; CoefficientList[Series[E^(x/(1 - 9*x^3)^(1/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, 9^k*binomial(n/3-1, k)/(n-3*k)!);
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1 - 9*x^3)^(1/3)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Sep 03 2025

Crossrefs

Cf. A189054, A371458, A373517.

Sequence in context: A142279 A092342 A170916 * A200908 A031418 A142229

Adjacent sequences: A386718 A386719 A386720 * A386722 A386723 A386724

Keyword

nonn

Author

Seiichi Manyama, Sep 03 2025

Status

approved