|
%I #9 Dec 23 2015 02:55:42
%S 1,3,4,7,11,12,29,15,49,43,100,100,157,45,299,159,273,795,761,307,830,
%T 2126,1657,3276,1711,965,3505,6405,1509,9967,6976,9375,8188,24483,
%U 8089,26299,20795,29871,40408,112475,51497,164022,27650,83398,74639,208015,280074
%N L.g.f.: Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + n*x^d/d).
%F Forms the logarithmic derivative of A205476.
%e L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 11*x^5/5 + 12*x^6/6 +...
%e By definition:
%e L(x) = x*(1+x) + x^2*(1+2*x)*(1+x^2)/2 + x^3*(1+3*x)*(1+x^3)/3 + x^4*(1+4*x)*(1+2*x^2)*(1+x^4)/4 + x^5*(1+5*x)*(1+x^5)/5 + x^6*(1+6*x)*(1+3*x^2)*(1+2*x^3)*(1+x^6)/6 +...
%e Exponentiation yields the g.f. of A205476:
%e exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 20*x^7 +...
%t max = 50; s = Sum[(x^(n-1)/n)*Product[1+n*x^d/d, {d, Divisors[n]}], {n, 1, max}] + O[x]^max; CoefficientList[s, x]*Range[max] (* _Jean-François Alcover_, Dec 23 2015 *)
%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. A205476 (exp), A205479, A205481, A205483, A205485, A205487, A205489, A205491.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 27 2012
|
