A351934 Expansion of e.g.f. exp(x / (1 - x^3/6)).

1, 1, 1, 1, 5, 21, 61, 281, 1961, 10585, 59641, 509521, 4086061, 29792621, 288051765, 2997900361, 27857436881, 302238265841, 3759751338481, 43054257392545, 526471319508181, 7463835257515141, 101755979383832621, 1404876681654178041, 22115830210157954425

Offset: 0

Author

Seiichi Manyama, Feb 26 2022

Keywords

• nonn

Crossrefs

Cf. A000262, A351933, A351935, A351936.

Cf. A293493.

Programs

Mathematica

m = 24; Range[0, m]! * CoefficientList[Series[Exp[x/(1 - x^3/6)], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^3/6))))

PARI

a(n) = if(n<4, 1, sum(k=0, (n-1)\3, (3*k+1)!/6^k*binomial(n-1, 3*k)*a(n-1-3*k)));

Formula

a(n) = Sum_{k=0..floor((n-1)/3)} (3*k+1)!/6^k * binomial(n-1,3*k) * a(n-1-3*k) for n > 3.

a(n) ~ n^(n - 1/4) / (2^(n/3 + 5/12) * 3^(n/3 + 1/6) * exp(n - 2^(7/6) * 3^(-1/3) * sqrt(n) - 6^(-2/3))). - Vaclav Kotesovec, Mar 03 2022

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k)/(6^k * (n-3*k)!). - Seiichi Manyama, Jun 08 2024