A198305 - OEIS

%I #7 Mar 30 2012 18:37:31

%S 1,3,7,19,51,159,519,1867,7234,30243,135125,642307,3231047,17138845,

%T 95554662,558384955,3411049542,21730279218,144048688538,991665854999,

%U 7077433997172,52283785492733,399238054300828,3147127294177099,25579801627862301,214139186144996635

%N  L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - n*x^d/d).

%C  Forms the logarithmic derivative of A198304.

%e  L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 51*x^5/5 + 159*x^6/6 +...

%e such that, by definition:

%e L(x) = x/(1-x) + (x^2/2)/((1-2*x)*(1-x^2)) + (x^3/3)/((1-3*x)*(1-x^3)) + (x^4/4)/((1-4*x)*(1-2*x^2)*(1-x^4)) + (x^5/5)/((1-5*x)*(1-x^5)) + (x^6/6)/((1-6*x)*(1-3*x^2)*(1-2*x^3)*(1-x^6)) +...+ (x^n/n)/Product_{d|n} (1-n*x^d/d) +...

%e Exponentiation yields the g.f. of A198304:

%e exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 54*x^6 + 148*x^7 +...

%o  (PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -log(1-m*x^d/d+x*O(x^n))))), n)}

%Y  Cf. A198304 (exp), A198299.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 27 2012