A336021 -id:A336021 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jul 05 2020

nonn

Shifts left 3 places under Stirling transform.

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).

Cf. A003659, A007469, A336021, A336022.

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

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]!

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

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

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

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 5, 15, 52, 203, 878, 4172, 21767, 125536, 809254, 5890115, 48560551, 450859572, 4657423009, 52802518648, 649162712358, 8574743501046, 120876064485660, 1809924607067234, 28694297293078915, 480719498205658859, 8502406681853097237

Offset: 0

Ilya Gutkovskiy, Jul 05 2020

nonn

Shifts left 5 places under Stirling transform.

Alois P. Heinz, Table of n, a(n) for n = 0..394

Cf. A003659, A007469, A336020, A336021.

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<5, 1, b(n-5, 0)):
seq(a(n), n=0..28);  # Alois P. Heinz, Aug 13 2021

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}]
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]!

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

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.

cp's OEIS Frontend

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

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