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

%I A246689 #31 Feb 25 2023 18:08:45
%S A246689 1,1,3,13,61,381,2791,22513,210393,2183401,24575851,305067621,
%T A246689 4097726293,58876485253,910581818511,15005958062761,261751577640241,
%U A246689 4844661893762193,94564968066402643,1938366513866527741,41760228574294689261,941821175462309114701
%N A246689 Expansion of e.g.f. 1/(1 - x^3)^(1 + 1/x + 1/x^2).
%C A246689 Compare with A193281.
%H A246689 Seiichi Manyama, <a href="/A246689/b246689.txt">Table of n, a(n) for n = 0..448</a>
%F A246689 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! + ....
%F A246689 A(x) = Sum_{n>=0} (x^n/n!)*Product {k = 1..n} (1 + x + k*x^2).
%F A246689 It appears that a(n) == 1 (mod n*(n-1)).
%F A246689 a(n) ~ n! * (n^2 / 54) * (1 + 6*log(n)/n). - _Vaclav Kotesovec_, Sep 01 2014
%F A246689 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
%p A246689 seq(coeftayl(n!/(1-x^3)^(1+1/x+1/x^2), x = 0, n), n = 0..10);
%t A246689 CoefficientList[Series[1/(1 - x^3)^(1 + 1/x + 1/x^2), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Sep 01 2014 *)
%o A246689 (PARI) my(x='x+O('x^66)); Vec(serlaplace(1/(1 - x^3)^(1 + 1/x + 1/x^2))) \\ _Joerg Arndt_, Sep 01 2014
%o A246689 (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
%Y A246689 Cf. A008620, A193281.
%K A246689 nonn,easy
%O A246689 0,3
%A A246689 _Peter Bala_, Sep 01 2014