A392889 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + (exp(x^2) - 1)/x) ).

1, 1, 2, 9, 72, 740, 9180, 137550, 2432640, 49354704, 1128304800, 28732944240, 807039858240, 24783163479360, 826010525340000, 29698116604556400, 1145780002586419200, 47217374768077113600, 2070009822252860167680, 96196395459475721329920, 4723635007762861462272000

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 + (exp((x*A(x))^2) - 1)/(x*A(x)).

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

Mathematica

Table[(n!)^2*Sum[1/(2*k+1)!*Abs[StirlingS2[n-k, n-2*k]/(n -k)!], {k, 0, Floor[n/2]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 27 2026 *)

Prog

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

Crossrefs

Cf. A000272, A392890.

Sequence in context: A381171 A370889 A367485 * A391838 A133941 A240956

Adjacent sequences: A392886 A392887 A392888 * A392890 A392891 A392892

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved