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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 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.)