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A349904 Inverse Euler transform of the tribonacci numbers A000073. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Table of n, a(n) for n=1..40.

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]]

PROG

(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)]))

(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)])

(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

CROSSREFS

Column k=2 of A349802.

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

Cf. A349903, A349977.

Sequence in context: A181532 A077930 A060945 * A023359 A082482 A066000

Adjacent sequences:  A349901 A349902 A349903 * A349905 A349906 A349907

KEYWORD

nonn

AUTHOR

Peter Luschny, Dec 05 2021

STATUS

approved

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Last modified August 9 03:22 EDT 2022. Contains 356016 sequences. (Running on oeis4.)