A376391 Expansion of e.g.f. ( (1/x) * Series_Reversion( x*(2 - exp(x))^3 ) )^(2/3).

1, 2, 20, 386, 11252, 441722, 21867764, 1308580226, 91904288420, 7413237414602, 675503178005108, 68631619821747842, 7693344955213551428, 943236099444038389082, 125565496331888560573172, 18037220418654308659836674, 2780985275750966018759898212, 458079154394191702424821932842

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Index entries for reversions of series

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367135.

a(n) = (2/(3*n+2)!) * Sum_{k=0..n} (3*n+k+1)! * Stirling2(n,k).

From Vaclav Kotesovec, Feb 03 2026: (Start)

a(n) ~ sqrt(1 + 1/(3 - 8*sqrt(s))) * 3^(n - 1/2) * s^(3*n/2 + 1) * (2*sqrt(s) - 1)^n * n^(n-1) / exp(n), where s = 1/(4*(1 - 1/(3*LambertW(2*exp(1/3)/3)))^2) = 1.692437741325114360587738... is the root of the equation exp(1/(6*sqrt(s)-3)) + 1/sqrt(s) = 2.

In closed form, a(n) ~ 3^(4*n+1) * LambertW(2*exp(1/3)/3)^(3*n+2) * n^(n-1) / (sqrt(1 + LambertW(2*exp(1/3)/3)) * 2^(3*n+1) * exp(n) * (3*LambertW(2*exp(1/3)/3) - 1)^(4*n+2)). (End)

Mathematica

Table[2/(3*n+2)! * Sum[(3*n+k+1)! * StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 03 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(2-exp(x))^3)/x)^(2/3)))
(PARI) a(n) = 2*sum(k=0, n, (3*n+k+1)!*stirling(n, k, 2))/(3*n+2)!;

Crossrefs

Cf. A367135, A376390.

Sequence in context: A379688 A187661 A263207 * A376394 A218306 A396509

Adjacent sequences: A376388 A376389 A376390 * A376392 A376393 A376394

Keyword

nonn

Author

Seiichi Manyama, Sep 22 2024

Status

approved