A391954 Expansion of e.g.f. Series_Reversion(log(1+x) - x^3).

1, 1, 7, 61, 751, 12181, 237007, 5489821, 146316751, 4421327461, 149295669007, 5571151966381, 227686268177551, 10113730235530741, 485169897369011407, 24997456057660894141, 1376733652387749583951, 80713621007842818156421, 5018684662081500344862607

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (3*k)!/k! * Stirling2(n+k,3*k+1).

E.g.f. A(x) satisfies A(x) = exp(x + A(x)^3) - 1.

E.g.f.: exp(B(x)) - 1, where B(x) is the e.g.f. of A392716.

a(n) ~ sqrt(1 + phi^(2/3)) * n^(n-1) / (sqrt(3) * 5^(1/4) * phi^(1/6) * exp(n) * ((1/phi^(1/3) - phi^(1/3))/3 + log((2 + 1/phi^(4/3) + phi^(4/3))/3))^(n - 1/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Feb 01 2026

Mathematica

Rest@CoefficientList[InverseSeries[Series[Log[1+x]-x^3, {x, 0, 20}]]//Normal, x]*Range[20]! (* Shenghui Yang, Jan 21 2026 *)
Table[Sum[(3*k)!/k!*StirlingS2[n+k, 3*k+1], {k, 0, Floor[(n-1)/2]}], {n, 1, 21}] (* Vincenzo Librandi, Feb 14 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(log(1+x)-x^3)))
(Magma) [ &+[Factorial(3*k)/Factorial(k)*StirlingSecond(n+k, 3*k+1): k in [0..Floor((n-1)/2)]]: n in [1..25]]; // Vincenzo Librandi, Feb 14 2026

Crossrefs

Cf. A143139, A392723.

Cf. A392716.

Sequence in context: A368324 A350157 A048287 * A317430 A145507 A254121

Adjacent sequences: A391951 A391952 A391953 * A391955 A391956 A391957

Keyword

nonn

Author

Seiichi Manyama, Jan 20 2026

Status

approved