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A336022 a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} Stirling2(n-5,k) * a(k).

%I A336022 #15 Aug 13 2021 21:30:20
%S A336022 1,1,1,1,1,1,1,2,5,15,52,203,878,4172,21767,125536,809254,5890115,
%T A336022 48560551,450859572,4657423009,52802518648,649162712358,8574743501046,
%U A336022 120876064485660,1809924607067234,28694297293078915,480719498205658859,8502406681853097237
%N A336022 a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} Stirling2(n-5,k) * a(k).
%C A336022 Shifts left 5 places under Stirling transform.
%H A336022 Alois P. Heinz, <a href="/A336022/b336022.txt">Table of n, a(n) for n = 0..394</a>
%F A336022 E.g.f. A(x) satisfies A(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + Integral( Integral( Integral( Integral( Integral A(exp(x) - 1) dx) dx) dx) dx) dx.
%p A336022 b:= proc(n, m) option remember; `if`(n=0,
%p A336022       a(m), m*b(n-1, m)+b(n-1, m+1))
%p A336022     end:
%p A336022 a:= n-> `if`(n<5, 1, b(n-5, 0)):
%p A336022 seq(a(n), n=0..28);  # _Alois P. Heinz_, Aug 13 2021
%t A336022 a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 5, k] a[k], {k, 0, n - 5}]; Table[a[n], {n, 0, 28}]
%t A336022 nmax = 28; A[_] = 0; Do[A[x_] = 1 + x + x^2/2 + x^3/6 + x^4/24 + Integrate[Integrate[Integrate[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
%o A336022 (PARI) lista(nn) = {my(va = vector(nn, k, 1)); for (n=6, nn, va[n] = sum(k=0, n-5, stirling(n-6, k, 2)*va[k+1]);); va;} \\ _Michel Marcus_, Jul 05 2020
%Y A336022 Cf. A003659, A007469, A336020, A336021.
%K A336022 nonn
%O A336022 0,8
%A A336022 _Ilya Gutkovskiy_, Jul 05 2020