A347002 - OEIS

%I #12 May 06 2022 20:33:45

%S 1,0,0,1,6,35,235,1834,16352,163764,1818030,22143726,293476326,

%T 4203311892,64682865156,1064154324024,18636296872320,346103784493560,

%U 6793394350116600,140508244952179200,3054120126193160280,69596730438090806880,1659041650323705102840

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

%H Seiichi Manyama, <a href="/A347002/b347002.txt">Table of n, a(n) for n = 0..447</a>

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

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

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

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

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

%Y Cf. A000399, A327504, A346922, A347001, A347003, A347004.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Aug 10 2021