%I #5 Jun 28 2021 15:50:02

%S 1,-3,12,-64,372,-2268,14394,-94296,632328,-4317846,29925108,

%T -209966748,1488507931,-10645680858,76717312932,-556528367791,

%U 4060765734816,-29782931545368,219444442931836,-1623585342758532,12057148232386980,-89842712017158526,671521130395037280

%N G.f. A(x) satisfies: A(x) = x / exp(3 * Sum_{k>=1} A(x^k) / k).

%F G.f.: x * Product_{n>=1} (1 - x^n)^(3*a(n)).

%F a(n+1) = -(3/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+1).

%p a:= proc(n) option remember; `if`(n=1, 1, -3*add(a(n-k)*

%p add(d*a(d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))

%p end:

%p seq(a(n), n=1..23); # _Alois P. Heinz_, Jun 28 2021

%t nmax = 23; A[_] = 0; Do[A[x_] = x/Exp[3 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

%t a[1] = 1; a[n_] := a[n] = -(3/(n - 1)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 23}]

%Y Cf. A006964, A045648, A052757, A345878.

%K sign

%O 1,2

%A _Ilya Gutkovskiy_, Jun 28 2021