OFFSET
0,3
COMMENTS
Logarithmic derivative yields A198299.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k ) ), where sigma(n,k) is the sum of the k-th powers of the divisors of n.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 17*x^6 + 22*x^7 +...
such that, by definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)) + (x^3/3)/((1-x^3)*(1-3*x^3)) + (x^4/4)/((1-x^4)*(1-2*x^4)*(1-4*x^4)) + (x^5/5)/((1-x^5)*(1-5*x^5)) + (x^6/6)/((1-x^6)*(1-2*x^6)*(1-3*x^6)*(1-6*x^6)) +...+ (x^n/n)/Product_{d|n} (1-d*x^n) +...
Also, we have the identity:
log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x
+ (1 + 3*x^2 + 7*x^4 + 15*x^6 + 31*x^8 +...)*x^2/2
+ (1 + 4*x^3 + 13*x^6 + 40*x^9 + 121*x^12 +...)*x^3/3
+ (1 + 7*x^4 + 35*x^8 + 155*x^12 + 651*x^16 +...)*x^4/4
+ (1 + 6*x^5 + 31*x^10 + 156*x^15 + 781*x^20 +...)*x^5/5
+ (1 + 12*x^6 + 97*x^12 + 672*x^18 + 4333*x^24 +...)*x^6/6 +...
+ exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k )*x^n/n +...
Explicitly, the logarithm begins:
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 36*x^6/6 + 8*x^7/7 + 83*x^8/8 + 49*x^9/9 + 178*x^10/10 +...+ A198299(n)*x^n/n +...
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sum(k=1, n\m, sigma(m, k)*x^(m*k)/k)+x*O(x^n)))), n)}
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -log(1-d*x^m+x*O(x^n)))))), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 26 2012
STATUS
approved