A392794 Expansion of e.g.f. (1/x) * Series_Reversion( x - (exp(x^2) - 1)^2 ).

1, 0, 0, 6, 0, 120, 2880, 2940, 403200, 8074080, 47174400, 3635866080, 72648576000, 1183289627520, 71843629777920, 1617274512453600, 49788393887232000, 2703881987142336000, 74947823184858624000, 3369543466000424993280, 176517590187551348736000

Offset

0,4

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)^2/x.

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

Mathematica

Table[(1/(n+1))* Sum[(2*(n-2*k))!/(n-2*k)!*(2*n-2*k)!*StirlingS2[n-k, 2*(n-2*k)]/(n-k)!, {k, 0, Floor[n/2]}], {n, 0, 21}] (* Vincenzo Librandi, Feb 12 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x-(exp(x^2)-1)^2)/x))
(Magma) [(1/(n+1)) * &+[Factorial(2*(n-2*k))/Factorial(n-2*k) * Factorial(2*n-2*k)* StirlingSecond(n-k, 2*(n-2*k)) / Factorial(n-k): k in [0..Floor(n/2)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 12 2026

Crossrefs

Cf. A392795, A392796.

Sequence in context: A052679 A392891 A392887 * A392791 A266218 A240818

Adjacent sequences: A392791 A392792 A392793 * A392795 A392796 A392797

Keyword

nonn

Author

Seiichi Manyama, Jan 22 2026

Status

approved