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Inverse Euler transform of the number of partitions in expanding space (A023881).
1

%I #5 Oct 09 2019 03:01:59

%S 1,2,9,67,625,7903,117649,2105342,43048905,1000976352,25937424601,

%T 743191207969,23298085122481,793763217701693,29192928060852217,

%U 1152939097060278256,48661191875666868481,2185919903971766191000

%N Inverse Euler transform of the number of partitions in expanding space (A023881).

%F a(n) = (1/n)*Sum_{d|n} sigma(d,d)*moebius(n/d).

%F a(n) ~ n^(n-1). - _Vaclav Kotesovec_, Oct 09 2019

%e Let G(x) = Sum_{n>=0} A023881(n)*x^n then

%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 82*x^4 + 725*x^5 + 8811*x^6 +...

%e G(x) = 1/[(1-x)*(1-x^2)^2*(1-x^3)^9*(1-x^4)^67*(1-x^5)^625*...].

%t Table[Sum[DivisorSigma[d, d]*MoebiusMu[n/d], {d, Divisors[n]}]/n, {n, 1, 20}] (* _Vaclav Kotesovec_, Oct 09 2019 *)

%o (PARI) {a(n)=(1/n)*sumdiv(n,d,sigma(d,d)*moebius(n/d))}

%Y Cf. A023881, A158947.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 31 2009