A384199 Expansion of e.g.f. log(1 + x)/(1 - 3*x).

0, 1, 5, 47, 558, 8394, 150972, 3171132, 76102128, 2054797776, 61643570400, 2034241452000, 73232652355200, 2856073920854400, 119955098448864000, 5397979517377171200, 259103015526429849600, 13214253812770712217600, 713569705533931031654400

Offset

0,3

Links

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

Formula

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

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

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

a(n) ~ log(4/3) * 3^n * n!. - Vaclav Kotesovec, May 23 2025

Mathematica

a[n_]:= n! * Sum[(-1)^(k-1)*3^(n-k)/k, {k, 1, n}]; Table[a[n], {n, 0, 18}] (* Vincenzo Librandi, May 22 2025 *)

Prog

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

Crossrefs

Cf. A024167, A383897.

Cf. A069015.

Sequence in context: A370100 A328032 A074192 * A058806 A302616 A006902

Adjacent sequences: A384196 A384197 A384198 * A384200 A384201 A384202

Keyword

nonn,easy

Author

Seiichi Manyama, May 22 2025

Status

approved