A384200 Expansion of e.g.f. -log(1 - 3*x)/(3 * (1 - x)).

0, 1, 5, 33, 294, 3414, 49644, 872388, 18001584, 426553776, 11408104800, 339766164000, 11148335337600, 399489448694400, 15520734764640000, 649782085752172800, 29160211264750540800, 1396381090351116441600, 71068392067688315596800, 3830710201119961857331200

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = n! * Sum_{k=1..n} 3^(k-1)/k.

a(n) = n * a(n-1) + 3^(n-1) * (n-1)!.

a(n) = (4*n-3) * a(n-1) - 3 * (n-1)^2 * a(n-2).

Mathematica

a[n_]:= n! * Sum[(3)^(k-1)/k, {k, 1, n}]; Table[a[n], {n, 0, 19}] (* Vincenzo Librandi, May 22 2025 *)

Prog

(PARI) a(n) = n!*sum(k=1, n, 3^(k-1)/k);
(Magma) [0] cat [n le 1 select 1  else  n * Self(n-1) + 3^(n-1) * Factorial(n-1): n in [1..20]]; // Vincenzo Librandi, May 22 2025

Crossrefs

Cf. A126674.

Sequence in context: A291846 A255927 A001828 * A084845 A198079 A098460

Adjacent sequences: A384197 A384198 A384199 * A384201 A384202 A384203

Keyword

nonn,easy

Author

Seiichi Manyama, May 22 2025

Status

approved