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A336020 a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} Stirling2(n-3,k) * a(k).

%I A336020 #17 Aug 13 2021 21:24:33
%S A336020 1,1,1,1,1,2,5,15,53,222,1115,6698,47243,382187,3480048,35251942,
%T A336020 394839407,4875966656,66282636371,989985346269,16198580140543,
%U A336020 289168351452220,5604120791540468,117309414122840454,2639927837211705159,63618153549702851338
%N A336020 a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} Stirling2(n-3,k) * a(k).
%C A336020 Shifts left 3 places under Stirling transform.
%H A336020 Michael De Vlieger, <a href="/A336020/b336020.txt">Table of n, a(n) for n = 0..371</a>
%H A336020 Ronald Orozco López, <a href="https://www.researchgate.net/publication/350397609_Solution_of_the_Differential_Equation_ykeay_Special_Values_of_Bell_Polynomials_and_ka-Autonomous_Coefficients">Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients</a>, Universidad de los Andes (Colombia 2021).
%F A336020 E.g.f. A(x) satisfies A(x) = 1 + x + x^2/2 + Integral( Integral( Integral A(exp(x) - 1) dx) dx) dx.
%p A336020 b:= proc(n, m) option remember; `if`(n=0,
%p A336020       a(m), m*b(n-1, m)+b(n-1, m+1))
%p A336020     end:
%p A336020 a:= n-> `if`(n<3, 1, b(n-3, 0)):
%p A336020 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 13 2021
%t A336020 a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 25}]
%t A336020 nmax = 25; A[_] = 0; Do[A[x_] = 1 + x + x^2/2 + Integrate[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
%o A336020 (PARI) lista(nn) = {my(va = vector(nn, k, 1)); for (n=4, nn, va[n] = sum(k=0, n-3, stirling(n-4, k, 2)*va[k+1]);); va;} \\ _Michel Marcus_, Jul 06 2020
%Y A336020 Cf. A003659, A007469, A336021, A336022.
%K A336020 nonn
%O A336020 0,6
%A A336020 _Ilya Gutkovskiy_, Jul 05 2020