A392888 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + log(1-x^3)^2/x^2) ).

1, 0, 0, 0, 24, 0, 0, 5040, 161280, 0, 3326400, 439084800, 10538035200, 5189184000, 1729036108800, 137305808640000, 2945115152179200, 10581700985856000, 1877496089204736000, 117959253866444390400, 2455507722669281280000, 32187293568176947200000

Offset

0,5

Links

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

Formula

E.g.f. A(x) satisfies A(x) = 1 + log(1-(x*A(x))^3)^2/(x*A(x))^2.

a(n) = (n!)^2 * Sum_{k=0..floor(n/3)} (2*(n-3*k))!/((n-3*k)! * (3*k+1)!) * |Stirling1(n-2*k,2*(n-3*k))|/(n-2*k)!.

Mathematica

Table[(n!)^2*Sum[(2*(n-3*k))!/((n-3*k)!*(3*k+1)!)*Abs[StirlingS1[n-2*k, 2*(n-3*k)]/(n -2*k)!], {k, 0, Floor[n/3]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 27 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+log(1-x^3)^2/x^2))/x))
(Magma) seq := [(Factorial(n))^2 * &+[Factorial(2*(n-3*k))/(Factorial(n-3*k) * Factorial(3*k+1)) * Abs(StirlingFirst(n - 2*k, 2*(n - 3*k)) / Factorial(n - 2*k)): k in [0..Floor(n/3)]] : n in [0..25] ]; seq; // Vincenzo Librandi, Jan 27 2026

Crossrefs

Cf. A392852, A392887.

Sequence in context: A376347 A376346 A392892 * A392795 A392792 A075406

Adjacent sequences: A392885 A392886 A392887 * A392889 A392890 A392891

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved