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

1, 1, 1, 1, 1, 2, 5, 15, 53, 222, 1115, 6698, 47243, 382187, 3480048, 35251942, 394839407, 4875966656, 66282636371, 989985346269, 16198580140543, 289168351452220, 5604120791540468, 117309414122840454, 2639927837211705159, 63618153549702851338

Offset: 0

Author

Ilya Gutkovskiy, Jul 05 2020

Keywords

• nonn

Comments

Shifts left 3 places under Stirling transform.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..371
• Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).

Crossrefs

Cf. A003659, A007469, A336021, 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<3, 1, b(n-3, 0)):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 13 2021

Mathematica

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}]
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]!

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

Formula

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