A246689 Expansion of e.g.f. 1/(1 - x^3)^(1 + 1/x + 1/x^2).

1, 1, 3, 13, 61, 381, 2791, 22513, 210393, 2183401, 24575851, 305067621, 4097726293, 58876485253, 910581818511, 15005958062761, 261751577640241, 4844661893762193, 94564968066402643, 1938366513866527741, 41760228574294689261, 941821175462309114701

Offset: 0

Author

Peter Bala, Sep 01 2014

Keywords

• nonn
• easy

Comments

Compare with A193281.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A008620, A193281.

Programs

Maple

seq(coeftayl(n!/(1-x^3)^(1+1/x+1/x^2), x = 0, n), n = 0..10);

Mathematica

CoefficientList[Series[1/(1 - x^3)^(1 + 1/x + 1/x^2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 01 2014 *)

PARI

my(x='x+O('x^66)); Vec(serlaplace(1/(1 - x^3)^(1 + 1/x + 1/x^2))) \\ Joerg Arndt, Sep 01 2014

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j/((j+2)\3)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Apr 30 2022

Formula

E.g.f.: A(x) = 1/(1 - x^3)^(1 + 1/x + 1/x^2) = exp( Sum_{n>=1} x^n/A008620(n-1) ) = 1 + x + 3*x^2/2! + 13*x^3/3! + 61*x^4/4! + ....

A(x) = Sum_{n>=0} (x^n/n!)*Product {k = 1..n} (1 + x + k*x^2).

It appears that a(n) == 1 (mod n*(n-1)).

a(n) ~ n! * (n^2 / 54) * (1 + 6*log(n)/n). - Vaclav Kotesovec, Sep 01 2014

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k/A008620(k-1) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 30 2022