A363116 Expansion of e.g.f. log(1 - (1/3)*log(1-3*x)).

0, 1, 2, 11, 93, 1068, 15486, 271206, 5566086, 130982328, 3476230344, 102709363392, 3343387479840, 118880973126576, 4584247231485312, 190548125567321328, 8492669888285758896, 404023626910206388224, 20434095445804056842112, 1094849162137482139541376

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.

(1) A(x) = log(1 - (1/3)*log(1-3*x)).

(2) a(n) = (-1)^(n-1) * Sum_{k=1..n} 3^(n-k) * (k-1)! * Stirling1(n, k) for n > 0.

(3) a(n) = 3^(n-1)*(n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * 3^(k-1) * a(n-k) for n > 0.

Example

E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 93*x^4/4! + 1068*x^5/5! + 15486*x^6/6! + 271206*x^7/7! + 5566086*x^8/8! + 130982328*x^9/9! + ...
where
exp(A(x)) = 1 + x + 3*x^2/2 + 9*x^3/3 + 27*x^4/4 + 81*x^5/5 + ... + 3^(n-1)*x^n/n + ...

Prog

(PARI) {a(n) = n!*polcoeff( log((1 - (1/3)*log(1-3*x +x*O(x^n) ))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = (-1)^(n-1) * sum(k=1, n, 3^(n-k) * (k-1)! * stirling(n, k, 1) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if (n<1, 0, 3^(n-1)*(n-1)! - sum(k=1, n-1, binomial(n-1, k)*(k-1)! * 3^(k-1) * a(n-k)))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A089064, A003713, A363115.

Sequence in context: A068392 A164893 A355111 * A099697 A346185 A083069

Adjacent sequences: A363113 A363114 A363115 * A363117 A363118 A363119

Keyword

nonn

Author

Paul D. Hanna, Jun 09 2023

Status

approved