A391838 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - log(1-x^2)/x) ).

1, 1, 2, 9, 72, 760, 9900, 156240, 2903040, 61943616, 1492545600, 40100860800, 1188788166720, 38545430489280, 1357015343523840, 51552166609944000, 2102033431456358400, 91568267279987097600, 4244171627969701908480, 208553808697356690247680, 10829949151457933392896000

Offset

0,3

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))^2)/(x*A(x)).

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

Mathematica

Table[(n!)^2* Sum[1/(2*k+1)!*Abs[StirlingS1[n-k, n-2*k]/(n-k)!], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vincenzo Librandi, Feb 05 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1-log(1-x^2)/x))/x))
(Magma) [Factorial(n)^2 * &+[1/Factorial(2*k+1) * Abs(StirlingFirst(n - k, n-2*k) / Factorial(n - k)): k in [0..Floor(n/2)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 05 2026

Crossrefs

Cf. A138013, A392886.

Sequence in context: A370889 A367485 A392889 * A133941 A240956 A038035

Adjacent sequences: A391835 A391836 A391837 * A391839 A391840 A391841

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved