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

%I #21 Feb 22 2023 23:13:20

%S 1,-1,-1,-1,13,19,-29,251,281,-13033,56071,-28601,-10136411,57321419,

%T -39757717,-17223709021,139901102641,-12418205969,-56710054724849,

%U 628073178260687,380303328920381,-324513582131326141,4616335903275095539,5642278545451902859

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

%H Vaclav Kotesovec, <a href="/A353205/b353205.txt">Table of n, a(n) for n = 0..449</a>

%F E.g.f.: exp( -Sum{k >= 1} x^k/A008620(k-1) ).

%F a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..n} k/A008620(k-1) * a(n-k)/(n-k)!.

%t nmax = 25; CoefficientList[Series[(1 - x^3)^(1 + 1/x + 1/x^2), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, May 09 2022 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)^(1+1/x+1/x^2)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, x^k/((k+2)\3)))))

%o (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;

%Y Cf. A008620, A246689, A353204.

%K sign

%O 0,5

%A _Seiichi Manyama_, Apr 30 2022