login
Expansion of e.g.f. exp(x^3/(6 * (1-x))).
6

%I #24 Jun 17 2024 08:49:52

%S 1,0,0,1,4,20,130,980,8400,80920,865200,10164000,130114600,1802600800,

%T 26867640800,428661633400,7288513232000,131558835408000,

%U 2512282795422400,50600743739145600,1071998968264224000,23829055696093648000,554524256514356128000

%N Expansion of e.g.f. exp(x^3/(6 * (1-x))).

%F a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,2) * a(n-3) - 2*binomial(n-1,3) * a(n-4) for n > 3.

%F a(n) ~ 2^(-3/4) * 3^(-1/4) * exp(-5/12 + sqrt(2*n/3) - n) * n^(n - 1/4). - _Vaclav Kotesovec_, Mar 29 2023

%F From _Seiichi Manyama_, Jun 17 2024: (Start)

%F a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/(6^k * k!).

%F a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)))))

%Y Cf. A185369, A361545, A361547.

%Y Cf. A293049.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Mar 15 2023