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A198299 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n). 4

%I

%S 1,3,4,11,6,36,8,83,49,178,12,680,14,920,714,2707,18,7119,20,14166,

%T 7844,22564,24,94616,3931,106538,88987,306604,30,832606,32,1401715,

%U 974736,2228278,150758,9643703,38,9961532,10363682,28802278,42,78793604,44,123016344

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

%C Forms the logarithmic derivative of A198296.

%H Paul D. Hanna, <a href="/A198299/b198299.txt">Table of n, a(n) for n = 1..500</a>

%F L.g.f.: 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.

%e L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 36*x^6/6 +...

%e such that, by definition:

%e L(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) +...

%e Also, we have the identity:

%e L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x

%e + (1 + 3*x^2 + 7*x^4 + 15*x^6 + 31*x^8 +...)*x^2/2

%e + (1 + 4*x^3 + 13*x^6 + 40*x^9 + 121*x^12 +...)*x^3/3

%e + (1 + 7*x^4 + 35*x^8 + 155*x^12 + 651*x^16 +...)*x^4/4

%e + (1 + 6*x^5 + 31*x^10 + 156*x^15 + 781*x^20 +...)*x^5/5

%e + (1 + 12*x^6 + 97*x^12 + 672*x^18 + 4333*x^24 +...)*x^6/6 +...

%e + exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k )*x^n/n +...

%e Exponentiation yields the g.f. of A198296:

%e exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 17*x^6 + 22*x^7 +...

%o (PARI) {a(n)=n*polcoeff(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)}

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

%Y Cf. A198296 (exp), A198305.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 26 2012

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Last modified September 22 09:44 EDT 2021. Contains 347606 sequences. (Running on oeis4.)