A392767 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x^2*(exp(x)-1)^2) ).

1, 0, 0, 0, 24, 120, 420, 1260, 205072, 4000752, 47197260, 432488100, 20160114744, 771242791176, 18929705779348, 351103866831900, 11370244305027360, 546028004495677152, 21432810874922389404, 656396603104088518740, 22741692767758350567880, 1126239056576866581907320

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (2*k)!/k! * (n+k)! * Stirling2(n-2*k,2*k)/(n-2*k)!.

Mathematica

Table[(1/(n+1))* Sum[(2*k)!/k!*(n+k)!*StirlingS2[n-2*k, 2*k]/(n-2*k)!, {k, 0, Floor[n/4]}], {n, 0, 21}] (* Vincenzo Librandi, Feb 13 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2*(exp(x)-1)^2))/x))
(Magma) [(1/(n+1)) * &+[Factorial(2*k) / Factorial(k) * Factorial(n+k)* StirlingSecond(n-2*k, 2*k)/Factorial(n-2*k): k in [0..Floor(n/4)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 13 2026

Crossrefs

Cf. A370989.

Sequence in context: A292979 A052760 A392827 * A179720 A235702 A052754

Adjacent sequences: A392764 A392765 A392766 * A392768 A392769 A392770

Keyword

nonn

Author

Seiichi Manyama, Jan 22 2026

Status

approved