A391377 - OEIS

%I #27 Feb 18 2026 04:02:07

%S 1,0,0,0,24,60,240,1260,88704,786240,7171200,71517600,2937876480,

%T 51372921600,807474769920,12722062953600,430490106224640,

%U 11120066371814400,256635966514636800,5726578592094796800,191258904944891289600,6233110401650436864000,189751022005164912230400

%N Expansion of e.g.f. 1/(1 + LambertW(x^3 * log(1-x))).

%C In general, if k >= 1 and e.g.f. = 1/(1 + LambertW(x^k * log(1-x))), then a(n) ~ c * n^n / (exp(n) * r^n), where r is the root of the equation r^k * log(1-r) = -exp(-1) and c = 1/(sqrt(k + exp(1)*r^(k+1)/(1-r))).

%H Vaclav Kotesovec, <a href="/A391377/b391377.txt">Table of n, a(n) for n = 0..420</a>

%F a(n) ~ c * n^n / (exp(n) * r^n), where r = 0.6837029856269721748413229435706375221748341873725... is the root of the equation r^3 * log(1-r) = -exp(-1) and c = 1/(sqrt(3 + exp(1)*r^4/(1-r))) = 0.4527767973401676021460856581728864911399167212456425...

%F a(n) = n! * Sum_{k=0..floor(n/4)} k^k * |Stirling1(n-3*k,k)|/(n-3*k)!. - _Seiichi Manyama_, Jan 26 2026

%t nmax = 25; CoefficientList[Series[1/(1 + LambertW[x^3*Log[1-x]]), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jan 25 2026 *)

%t Join[{1},Table[n!* Sum[(k^k)*Abs[StirlingS1[n-3*k,k]]/(n-3*k)!,{k,1,Floor[n/4]}],{n,1,25}]] (* _Vincenzo Librandi_, Feb 04 2026 *)

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1 + lambertw(x^3 * log(1-x))))) \\ _Michel Marcus_, Jan 25 2026

%o (Magma) [Factorial(n)*&+[k^k* Abs(StirlingFirst(n-3*k,k))/Factorial(n-3*k): k in [0..Floor(n/4)] ] : n in [0..24] ]; // _Vincenzo Librandi_, Feb 04 2026

%Y Cf. A305981, A362835, A362891.

%K nonn

%O 0,5

%A _Vaclav Kotesovec_, Jan 25 2026