OFFSET
1,2
COMMENTS
In general, if p > q > 0 and e.g.f. = Series_Reversion(p*x + log(1-q*x)), then a(n) ~ n^(n-1) / (p * exp(n) * (p/q - log(p/q) - 1)^(n - 1/2)). - Vaclav Kotesovec, Jan 20 2026
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..250
FORMULA
G.f.: (x/4) * Sum_{k>=0} Product_{j=0..k-1} ((3/4) * (1 + j*x)).
a(1) = 1; a(n) = -3*a(n-1) + 6*Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k).
a(n) = (1/3) * Sum_{k>=0} (3/4)^(n+k) * |Stirling1(n-1+k,k)|.
a(n) ~ n^(n-1) / (4 * exp(n) * (1/3 - log(4/3))^(n - 1/2)). - Vaclav Kotesovec, Jan 19 2026
E.g.f.: 1/3 + LambertW(-1, -4*exp(x - 4/3)/3)/4. - Vaclav Kotesovec, Jan 20 2026
MATHEMATICA
numTerms=18; v={1}; Do[ v=Append[v, -3 v[[-1]] +6*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[4*x + Log[1 - 3*x], {x, 0, nmax}], x], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Jan 19 2026 *)
nmax = 20; Rest[Assuming[{x > 0}, CoefficientList[Series[1/3 + LambertW[-1, -4*E^(x - 4/3)/3]/4, {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(4*x+log(1-3*x))))
(Magma) N := 18; v := [1]; for n in [1..N-1] do Append(~v, -3*v[n] + 6*&+[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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 23 2025
STATUS
approved