OFFSET
0,4
FORMULA
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * mu(k) * a(n-k)/(n-k)!.
E.g.f. A(x) satisfies Product_{n>=1} A(x^n) = exp(x). - Paul D. Hanna, Feb 29 2024
EXAMPLE
E.g.f.: A(x) = 1 + x - x^2/2! - 11*x^3/3! - 23*x^4/4! - 19*x^5/5! + 991*x^6/6! + 4369*x^7/7! - 11311*x^8/8! - 356903*x^9/9! + 5389471*x^10/10! + ...
where A(x) = exp(x - x^2 - x^3 - x^5 + x^6 - x^7 + ... + mu(n)*x^n +....);
thus, exp(x) = A(x) * A(x^2) * A(x^3) * ... * A(x^n) * ...
MATHEMATICA
nmax = 20; A[_] = 1; Do[A[x_] = Exp[x]/Product[A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]*Range[0, nmax]! (* Vaclav Kotesovec, Mar 01 2024 *)
PROG
(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, moebius(k)*x^k))))
(PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*moebius(k)*a(n-k)/(n-k)!));
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Apr 06 2022
STATUS
approved