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 A205814 G.f.: Product_{n>=1} [ (1 - 2^n*x^n) / (1 - (n+2)^n*x^n) ]^(1/n). 7

%I

%S 1,1,9,54,482,4239,55561,785554,14133055,285547760,6666380256,

%T 172748192767,4974178683908,156462697434990,5354832107694444,

%U 197710292330150160,7839473395324929677,332071887435037103895,14968498613432649146050,715294449027151380463781

%N G.f.: Product_{n>=1} [ (1 - 2^n*x^n) / (1 - (n+2)^n*x^n) ]^(1/n).

%C Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

%H Vaclav Kotesovec, <a href="/A205814/b205814.txt">Table of n, a(n) for n = 0..380</a>

%F G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 2^(n-k) ).

%F a(n) ~ exp(2) * n^(n-1). - _Vaclav Kotesovec_, Oct 08 2016

%e G.f.: A(x) = 1 + x + 9*x^2 + 54*x^3 + 482*x^4 + 4239*x^5 + 55561*x^6 +...

%e where the g.f. equals the product:

%e A(x) = (1-2*x)/(1-3*x) * ((1-2^2*x^2)/(1-4^2*x^2))^(1/2) * ((1-2^3*x^3)/(1-5^3*x^3))^(1/3) * ((1-2^4*x^4)/(1-6^4*x^4))^(1/4) * ((1-2^5*x^5)/(1-7^5*x^5))^(1/5) *...

%e The logarithm equals the l.g.f. of A205815:

%e log(A(x)) = x + 17*x^2/2 + 136*x^3/3 + 1585*x^4/4 + 16986*x^5/5 +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*2^(m-k))+x*O(x^n))), n)}

%o (PARI) {a(n)=polcoeff(prod(k=1, n, ((1-2^k*x^k)/(1-(k+2)^k*x^k +x*O(x^n)))^(1/k)), n)}

%Y Cf. A205815 (log), A205811, A023881.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 01 2012

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Last modified November 27 19:48 EST 2022. Contains 358406 sequences. (Running on oeis4.)