A158267 - OEIS

%I #4 Oct 09 2019 02:34:37

%S 1,4,13,59,151,916,1961,12035,35110,166204,384781,3154367,5600323,

%T 34384676,124093963,582290595,1235438587,9831378712,18602770421,

%U 144738772109,410101237013,1721535323380,4295702988313,40309503022439

%N Inverse Euler transform of A156305.

%C G.f. of A156305: exp( Sum_{n>=1} sigma(n)*C(2*n-1,n)*x^n/n ), where C(2n-1,n) = A001700(n-1).

%F a(n) = (1/n)*Sum_{d|n} sigma(d)*C(2d-1,d)*moebius(n/d).

%e Let G(x) = g.f. of A156305:

%e G(x) = 1 + x + 5*x^2 + 18*x^3 + 87*x^4 + 290*x^5 + 1553*x^6 +...

%e G(x) = 1/[(1-x)*(1-x^2)^4*(1-x^3)^13*(1-x^4)^59*(1-x^5)^151*...].

%t Table[Sum[DivisorSigma[1, d]*Binomial[2*d - 1, d]*MoebiusMu[n/d], {d, Divisors[n]}] / n, {n, 1, 30}] (* _Vaclav Kotesovec_, Oct 09 2019 *)

%o (PARI) {a(n)=(1/n)*sumdiv(n,d, sigma(d)*binomial(2*d-1, d)*moebius(n/d))}

%Y Cf. A156305, A001700.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Apr 09 2009