A352868 Expansion of e.g.f. exp(Sum_{k>=1} mu(k) * x^k), where mu() is the Moebius function (A008683).
1, 1, -1, -11, -23, -19, 991, 4369, -11311, -356903, 5389471, 7875341, -430708871, -16579950971, 45417621887, 3629980647721, 93982540029601, -1077931879771471, -29167938898699841, -486520057714400603, 7973931691642326281, 205214099791890382621
Offset: 0
Keywords
Examples
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) * ...
Programs
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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 *)
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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)!));
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