A345232 G.f. A(x) satisfies: A(x) = x + x^2 / exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

1, 1, -1, -1, 2, 0, -4, 3, 7, -12, -8, 35, -6, -87, 84, 172, -367, -187, 1175, -417, -3003, 3621, 5723, -15126, -4374, 47813, -26192, -119731, 175835, 211797, -699210, -57982, 2148031, -1601079, -5161935, 9125489, 8093890, -34478125, 3997517, 101971205, -97182026

Offset

1,5

Links

Table of n, a(n) for n=1..41.

Formula

G.f.: x + x^2 * Product_{n>=1} (1 - x^n)^a(n).

a(n+2) = -(1/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).

Maple

a:= proc(n) option remember; `if`(n<3, 1, -add(add(a(n-k)*
      d*a(d), d=numtheory[divisors](k)), k=1..n-2)/(n-2))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Jun 11 2021

Mathematica

nmax = 41; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[-A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; a[n_] := a[n] = -(1/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 41}]

Crossrefs

Cf. A007562, A045648, A345233.

Sequence in context: A243488 A154849 A339261 * A277333 A248663 A335426

Adjacent sequences: A345229 A345230 A345231 * A345233 A345234 A345235

Keyword

sign

Author

Ilya Gutkovskiy, Jun 11 2021

Status

approved