This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363116 #5 Jun 09 2023 08:44:04
%S A363116 0,1,2,11,93,1068,15486,271206,5566086,130982328,3476230344,
%T A363116 102709363392,3343387479840,118880973126576,4584247231485312,
%U A363116 190548125567321328,8492669888285758896,404023626910206388224,20434095445804056842112,1094849162137482139541376
%N A363116 Expansion of e.g.f. log(1 - (1/3)*log(1-3*x)).
%F A363116 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
%F A363116 (1) A(x) = log(1 - (1/3)*log(1-3*x)).
%F A363116 (2) a(n) = (-1)^(n-1) * Sum_{k=1..n} 3^(n-k) * (k-1)! * Stirling1(n, k) for n > 0.
%F A363116 (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.
%e A363116 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! + ...
%e A363116 where
%e A363116 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 + ...
%o A363116 (PARI) {a(n) = n!*polcoeff( log((1 - (1/3)*log(1-3*x +x*O(x^n) ))),n)}
%o A363116 for(n=0,20,print1(a(n),", "))
%o A363116 (PARI) {a(n) = (-1)^(n-1) * sum(k=1,n, 3^(n-k) * (k-1)! * stirling(n, k, 1) )}
%o A363116 for(n=0,20,print1(a(n),", "))
%o A363116 (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)))}
%o A363116 for(n=0,20,print1(a(n),", "))
%Y A363116 Cf. A089064, A003713, A363115.
%K A363116 nonn
%O A363116 0,3
%A A363116 _Paul D. Hanna_, Jun 09 2023