A185301 - OEIS

%I #11 Mar 30 2012 18:37:25

%S 1,1,3,5,14,20,59,83,229,350,878,1302,3479,5068,12640,19357,47547,

%T 71173,175029,262445,635297,966680,2288213,3470143,8266788,12507003,

%U 29283071,44756825,104067224,158535387,367088494,559952784,1287857188,1971948577

%N G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n)^k * x^(n*k) / n ).

%F Logarithmic derivative yields A185302.

%e G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 14*x^4 + 20*x^5 + 59*x^6 + 83*x^7 +...

%e such that, by definition:

%e log(A(x)) = (x + x^2 + x^3 + x^4 +...+ x^k +...)/1

%e  + (3*x^2 + 9*x^4 + 27*x^6 + 81*x^8 +...+ 3^k*x^(2*k) +...)/2

%e  + (4*x^3 + 16*x^6 + 64*x^9 + 256*x^12 +...+ 4^k*x^(3*k) +...)/3

%e  + (7*x^4 + 49*x^8 + 343*x^12 + 2401*x^16 +...+ 7^k*x^(4*k) +...)/4 +...

%e = x + 5*x^2/2 + 7*x^3/3 + 29*x^4/4 + 11*x^5/5 + 131*x^6/6 +...+ A185302(n)*x^n/n +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=1, n\m, sigma(m)^k*x^(m*k)/m),x*O(x^n))), n)}

%Y Cf. A185302.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 25 2012