A213641 E.g.f. A(x) satisfies A(x) = 1 - log(1 - x^2*A(x)^2) / x.

1, 1, 4, 33, 408, 6760, 140880, 3543120, 104469120, 3535037856, 135053291520, 5750579640960, 270067321117440, 13868724577593600, 773138898730598400, 46500352460941579200, 3001412657729335449600, 206946807350480534937600, 15180752044039172426035200

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

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

E.g.f.: A(x) = (1/x)*Series_Reversion(x + log(1-x^2)).

a(n) = A213640(n+1)/(n+1).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (2*n-2*k)! * |Stirling1(n-k,n-2*k)|/(n-k)!. - Seiichi Manyama, Jan 23 2026

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! + 6760*x^5/5! +...
Related expansions:
A(x)^2 = 1 + 2*x + 10*x^2/2! + 90*x^3/3! + 1176*x^4/4! + 20240*x^5/5! +...
-log(1 - x^2*A(x)^2)/x = x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! +...
A(x + log(1-x^2)) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 340*x^5/5! +...

Prog

(PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x+log(1-x^2 +x^2*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A213640, A218653, A200320.

Sequence in context: A215364 A370930 A392788 * A343673 A343686 A364980

Adjacent sequences: A213638 A213639 A213640 * A213642 A213643 A213644

Keyword

nonn

Author

Paul D. Hanna, Jun 17 2012

Status

approved