A390902 - OEIS

%I #25 Jan 20 2026 05:47:27

%S 1,9,297,16281,1248777,123125913,14836317129,2112675966297,

%T 347116504354569,64634936370634521,13451125114945088073,

%U 3093938638419767225049,779418017792495272404873,213421992338103533527717017,63114617040470440017309335241,20047195073135198062149574564953

%N Expansion of e.g.f. Series_Reversion(4*x + log(1-3*x)).

%C 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

%H Vincenzo Librandi, <a href="/A390902/b390902.txt">Table of n, a(n) for n = 1..250</a>

%F G.f.: (x/4) * Sum_{k>=0} Product_{j=0..k-1} ((3/4) * (1 + j*x)).

%F a(1) = 1; a(n) = -3*a(n-1) + 6*Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k).

%F a(n) = (1/3) * Sum_{k>=0} (3/4)^(n+k) * |Stirling1(n-1+k,k)|.

%F a(n) ~ n^(n-1) / (4 * exp(n) * (1/3 - log(4/3))^(n - 1/2)). - _Vaclav Kotesovec_, Jan 19 2026

%F E.g.f.: 1/3 + LambertW(-1, -4*exp(x - 4/3)/3)/4. - _Vaclav Kotesovec_, Jan 20 2026

%t 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 *)

%t 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 *)

%t 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 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(4*x+log(1-3*x))))

%o (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

%Y Cf. A032188, A390901.

%Y Cf. A390900.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Nov 23 2025