A345883 G.f. A(x) satisfies: A(x) = x / exp(3 * Sum_{k>=1} A(x^k) / k).

1, -3, 12, -64, 372, -2268, 14394, -94296, 632328, -4317846, 29925108, -209966748, 1488507931, -10645680858, 76717312932, -556528367791, 4060765734816, -29782931545368, 219444442931836, -1623585342758532, 12057148232386980, -89842712017158526, 671521130395037280

Offset: 1

Author

Ilya Gutkovskiy, Jun 28 2021

Keywords

• sign

Crossrefs

Cf. A006964, A045648, A052757, A345878.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 1, -3*add(a(n-k)*
      add(d*a(d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Jun 28 2021

Mathematica

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
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}]

Formula

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

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