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 A198304 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - n*x^d/d) ). 3

%I

%S 1,1,2,4,9,21,54,148,442,1433,5061,19394,80308,357241,1697870,8577240,

%T 45845235,258198133,1526631800,9445795717,60988643813,409933740177,

%U 2862338202947,20723903238290,155329193200741,1203428108558453,9624564394649845,79357873429159078

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

%C Logarithmic derivative yields A198305.

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 54*x^6 + 148*x^7 +...

%e such that, by definition:

%e log(A(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 Explicitly, the logarithm begins:

%e log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 51*x^5/5 + 159*x^6/6 + 519*x^7/7 + 1867*x^8/8 + 7234*x^9/9 +...+ A198305(n)*x^n/n +...

%o (PARI) {a(n)=polcoeff(exp(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. A198305 (log), A198296.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 27 2012

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Last modified December 5 14:32 EST 2021. Contains 349557 sequences. (Running on oeis4.)