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

Author

Seiichi Manyama, May 22 2025

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A126674.

Programs

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

Mathematica

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

PARI

a(n) = n!*sum(k=1, n, 3^(k-1)/k);

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).