A390901 Expansion of e.g.f. Series_Reversion(3*x + log(1-2*x)).

1, 4, 64, 1696, 62848, 2993152, 174195712, 11979931648, 950590849024, 85481882484736, 8591082951540736, 954287803965374464, 116094972360232271872, 15351651107135845040128, 2192385904380293863702528, 336287208602252678237519872, 55139793645260833680781213696

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..250

Formula

G.f.: (x/3) * Sum_{k>=0} Product_{j=0..k-1} ((2/3) * (1 + j*x)).

a(1) = 1; a(n) = -2*a(n-1) + 3*Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k).

a(n) = (1/2) * Sum_{k>=0} (2/3)^(n+k) * |Stirling1(n-1+k,k)|.

a(n) ~ n^(n-1) / (3 * exp(n) * (1/2 - log(3/2))^(n - 1/2)). - Vaclav Kotesovec, Jan 19 2026

E.g.f.: 1/2 + LambertW(-1, -3*exp(x - 3/2)/2)/3. - Vaclav Kotesovec, Jan 20 2026

Mathematica

numTerms=18; v={1}; Do[ v=Append[v, -2 v[[-1]] +3 Sum[Binomial[n+1, k+1] v[[k+1]] v[[n-k]], {k, 0, n-1}]], {n, numTerms-1}]; v (* Vincenzo Librandi, Jan 19 2026 *)
nmax = 20; Rest[CoefficientList[InverseSeries[Series[3*x + Log[1 - 2*x], {x, 0, nmax}], x], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Jan 19 2026 *)
nmax = 20; Rest[Assuming[{x > 0}, CoefficientList[Series[1/2 + LambertW[-1, -3*E^(x - 3/2)/2]/3, {x, 0, nmax}], x] * Range[0, nmax]!]] (* Vaclav Kotesovec, Jan 20 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(3*x+log(1-2*x))))
(Magma) N := 18; v := [1]; for n in [1..N-1] do Append(~v, -2*v[n] + 3*&+[Binomial(n+1, k+1)*v[k+1]*v[n-k] : k in [0..n-1]]); end for; v; // Vincenzo Librandi, Jan 19 2026

Crossrefs

Cf. A032188, A390902.

Cf. A390899.

Sequence in context: A352275 A146341 A264335 * A091483 A263988 A208357

Adjacent sequences: A390898 A390899 A390900 * A390902 A390903 A390904

Keyword

nonn

Author

Seiichi Manyama, Nov 23 2025

Status

approved