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
Keywords
Examples
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 + ...
Programs
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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),", "))
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.