A336021 a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} Stirling2(n-4,k) * a(k).

1, 1, 1, 1, 1, 1, 2, 5, 15, 52, 204, 902, 4532, 26196, 175320, 1351296, 11819348, 115309534, 1236465988, 14419850138, 181652022376, 2462053028798, 35834756184146, 559816444117400, 9389648056139010, 169166236946379128, 3273760080403458226, 67994123544008546820

Offset: 0

Author

Ilya Gutkovskiy, Jul 05 2020

Keywords

• nonn

Comments

Shifts left 4 places under Stirling transform.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..384

Crossrefs

Cf. A003659, A007469, A336020, A336022.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0,
      a(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> `if`(n<4, 1, b(n-4, 0)):
seq(a(n), n=0..27);  # Alois P. Heinz, Aug 13 2021

Mathematica

a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 4, k] a[k], {k, 0, n - 4}]; Table[a[n], {n, 0, 27}]
nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2/2 + x^3/6 + Integrate[Integrate[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!

PARI

lista(nn) = {my(va = vector(nn, k, 1)); for (n=5, nn, va[n] = sum(k=0, n-4, stirling(n-5, k, 2)*va[k+1]);); va;} \\ Michel Marcus, Jul 06 2020

Formula

E.g.f. A(x) satisfies A(x) = 1 + x + x^2/2 + x^3/6 + Integral( Integral( Integral( Integral A(exp(x) - 1) dx) dx) dx) dx.