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A349904 Inverse Euler transform of the tribonacci numbers A000073.

%I A349904 #19 Dec 07 2021 18:16:54
%S A349904 0,0,1,1,2,3,6,10,18,31,56,96,172,299,530,929,1646,2893,5126,9044,
%T A349904 16028,28362,50328,89249,158598,281830,501538,892857,1591282,2837467,
%U A349904 5064334,9044023,16163946,28906213,51729844,92628401,165967884,297541263,533731692,957921314
%N A349904 Inverse Euler transform of the tribonacci numbers A000073.
%p A349904 read transforms;  # https://oeis.org/transforms.txt
%p A349904 arow := len -> EULERi([seq(A000073(n), n = 0..len)]): arow(39);
%p A349904 # second Maple program:
%p A349904 t:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1, 3]:
%p A349904 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A349904       add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p A349904     end:
%p A349904 a:= proc(n) option remember; t(n-1)-b(n, n-1) end:
%p A349904 seq(a(n), n=1..40);  # _Alois P. Heinz_, Dec 05 2021
%t A349904 (* EulerInvTransform is defined in A022562. *)
%t A349904 EulerInvTransform[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 40]]
%o A349904 (SageMath)
%o A349904 def euler_invtrans(A) :
%o A349904     L = []; M = []
%o A349904     for i in range(len(A)) :
%o A349904         s = (i+1)*A[i] - sum(L[j-1]*A[i-j] for j in (1..i))
%o A349904         L.append(s)
%o A349904         s = sum(moebius((i+1)/d)*L[d-1] for d in divisors(i+1))
%o A349904         M.append(s/(i + 1))
%o A349904     return M
%o A349904 @cached_function
%o A349904 def a(n): return a(n-1) + a(n-2) + a(n-3) if n > 2 else [0,0,1][n]
%o A349904 print(euler_invtrans([a(n) for n in range(40)]))
%o A349904 (Python)  # After the Maple program of _Alois P. Heinz_.
%o A349904 from functools import cache
%o A349904 from math import comb
%o A349904 def binomial(n, k):
%o A349904     if n == -1: return 1
%o A349904     return comb(n, k)
%o A349904 @cache
%o A349904 def A000073(n):
%o A349904     if n <= 1: return 0
%o A349904     if n == 2: return 1
%o A349904     return A000073(n-1) + A000073(n-2) + A000073(n-3)
%o A349904 @cache
%o A349904 def b(n, i):
%o A349904     if n == 0: return 1
%o A349904     if i <  1: return 0
%o A349904     return sum(binomial(a(i) + j - 1, j) *
%o A349904                b(n - i * j, i - 1) for j in range(1 + n // i))
%o A349904 @cache
%o A349904 def a(n): return (A000073(n - 1) - b(n, n - 1))
%o A349904 print([a(n) for n in range(1, 41)])
%o A349904 (PARI)
%o A349904 InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
%o A349904 seq(n) = InvEulerT(Vec(x^2/(1 - x - x^2 - x^3) + O(x^n), -n)) \\ _Andrew Howroyd_, Dec 05 2021
%Y A349904 Column k=2 of A349802.
%Y A349904 Cf. A000073, A057597 (tribonacci numbers for n <= 0), A006206 and A060280.
%Y A349904 Cf. A349903, A349977.
%K A349904 nonn
%O A349904 1,5
%A A349904 _Peter Luschny_, Dec 05 2021