A392762 E.g.f. A(x) satisfies A(x) = 1/(1 - log(1-x*A(x)^2)^2).

1, 0, 2, 6, 142, 1540, 40028, 830088, 26194320, 803816928, 30965575632, 1243606193280, 57518659025232, 2831567070616128, 154016101401783840, 8940598021282475520, 561024734955178219392, 37443206368654913645568, 2666651445545095799647488, 200981356103135218506923520

Offset

0,3

Links

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

Formula

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

a(n) ~ sqrt((s-1)*s^(5/2)/(sqrt(s-1) + sqrt(s)*(6*s - 5))) * (s^2 + 4*sqrt(s-1)*s^(5/2))^n * n^(n-1) / exp(n), where s = 1.2022277155248881869907262842781... is the root of the equation 4*(exp(sqrt(1 - 1/s)) - 1) * sqrt(s*(s-1)) = 1. - Vaclav Kotesovec, Jan 22 2026

Mathematica

Table[1/(2*n+1)! * Sum[(2*k)! / k! * (2*n+k)! * Abs[StirlingS1[n, 2*k]], {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 22 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (2*k)!/k!*(2*n+k)!*abs(stirling(n, 2*k, 1)))/(2*n+1)!;
(Magma) [(1/Factorial(2*n+1)) * &+[Factorial(2*k) / Factorial(k) * Factorial(2*n+k)* Abs(StirlingFirst(n, 2*k)): k in [0..Floor(n/2)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 13 2026

Crossrefs

Cf. A392768.

Sequence in context: A090907 A159478 A047937 * A027731 A280821 A145143

Adjacent sequences: A392759 A392760 A392761 * A392763 A392764 A392765

Keyword

nonn

Author

Seiichi Manyama, Jan 21 2026

Status

approved