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