A349904 Inverse Euler transform of the tribonacci numbers A000073.

0, 0, 1, 1, 2, 3, 6, 10, 18, 31, 56, 96, 172, 299, 530, 929, 1646, 2893, 5126, 9044, 16028, 28362, 50328, 89249, 158598, 281830, 501538, 892857, 1591282, 2837467, 5064334, 9044023, 16163946, 28906213, 51729844, 92628401, 165967884, 297541263, 533731692, 957921314

Offset: 1

Author

Peter Luschny, Dec 05 2021

Keywords

• nonn

Crossrefs

Column k=2 of A349802.

Cf. A000073, A057597 (tribonacci numbers for n <= 0), A006206 and A060280.

Cf. A349903, A349977.

Programs

Maple

read transforms;  # https://oeis.org/transforms.txt
arow := len -> EULERi([seq(A000073(n), n = 0..len)]): arow(39);
# second Maple program:
t:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1, 3]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; t(n-1)-b(n, n-1) end:
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 05 2021

Mathematica

(* EulerInvTransform is defined in A022562. *)
EulerInvTransform[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 40]]

PARI

InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n) = InvEulerT(Vec(x^2/(1 - x - x^2 - x^3) + O(x^n), -n)) \\ Andrew Howroyd, Dec 05 2021

Python

# After the Maple program of Alois P. Heinz.
from functools import cache
from math import comb
def binomial(n, k):
    if n == -1: return 1
    return comb(n, k)
@cache
def A000073(n):
    if n <= 1: return 0
    if n == 2: return 1
    return A000073(n-1) + A000073(n-2) + A000073(n-3)
@cache
def b(n, i):
    if n == 0: return 1
    if i <  1: return 0
    return sum(binomial(a(i) + j - 1, j) *
               b(n - i * j, i - 1) for j in range(1 + n // i))
@cache
def a(n): return (A000073(n - 1) - b(n, n - 1))
print([a(n) for n in range(1, 41)])

SageMath

def euler_invtrans(A) :
    L = []; M = []
    for i in range(len(A)) :
        s = (i+1)*A[i] - sum(L[j-1]*A[i-j] for j in (1..i))
        L.append(s)
        s = sum(moebius((i+1)/d)*L[d-1] for d in divisors(i+1))
        M.append(s/(i + 1))
    return M
@cached_function
def a(n): return a(n-1) + a(n-2) + a(n-3) if n > 2 else [0,0,1][n]
print(euler_invtrans([a(n) for n in range(40)]))