A346922 - OEIS

%I #22 May 09 2022 08:38:16

%S 1,0,0,1,6,35,245,2044,19572,210524,2513760,33012276,472963876,

%T 7340889192,122703087416,2197496734224,41979155247520,852063971170960,

%U 18312093589455440,415420659953439840,9920128280950954080,248735658391768241280,6533773435848445617600

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

%H Seiichi Manyama, <a href="/A346922/b346922.txt">Table of n, a(n) for n = 0..437</a>

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).

%F a(n) ~ n! * 6^(1/3) / (3 * exp(6^(1/3)) * (1 - exp(-6^(1/3)))^(n+1)). - _Vaclav Kotesovec_, Aug 08 2021

%F a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/6^k. - _Seiichi Manyama_, May 06 2022

%t nmax = 22; CoefficientList[Series[1/(1 + Log[1 - x]^3/3!), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 3]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-x)^3/3!))) \\ _Michel Marcus_, Aug 07 2021

%o (PARI) a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/6^k); \\ _Seiichi Manyama_, May 06 2022

%Y Cf. A007840, A346921, A346923, A346924.

%Y Cf. A000399, A346894, A347002, A353118.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Aug 07 2021