_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jan 27 2012

proposed

approved

editing

proposed

allocated for Paul D. Hanna

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

1, 1, 2, 4, 10, 22, 57, 134, 331, 797, 1995, 4879, 12367, 31056, 79315, 202370, 521575, 1339934, 3456778, 8885907, 22848211, 58576714, 150117209, 384135566, 983789032, 2522109065, 6485104365, 16736092434, 43408268497, 113201300205, 296975753940, 783578962587

0,3

Note: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - x^d)^n ) does not yield an integer series.

Logarithmic derivative yields A205491.

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 57*x^6 + 134*x^7 +...

By definition:

log(A(x)) = x/(1-x) + (x^2/2)/((1-x)^2*(1-2*x^2)^2) + (x^3/3)/((1-x)^3*(1-3*x^3)^3) + (x^4/4)/((1-x)^4*(1-2*x^2)^4*(1-4*x^4)^4) + (x^5/5)/((1-x)^5*(1-5*x^5)^5) + (x^6/6)/((1-x)^6*(1-2*x^2)^6*(1-3*x^3)^6*(1-6*x^6)^6) +...

Explicitly,

log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 51*x^5/5 + 165*x^6/6 + 386*x^7/7 + 1039*x^8/8 + 2554*x^9/9 +...+ A205491(n)*x^n/n +...

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

Cf. A205491 (log), A205476, A205478, A205480, A205482, A205484, A205486, A205488.

allocated

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 27 2012

approved

editing

allocated for Paul D. Hanna

allocated

approved