%I #23 Mar 20 2017 11:24:22

%S 1,12,86,469,2141,8594,31247,104945,330094,982284,2786861,7584060,

%T 19893185,50494558,124437410,298555264,699017259,1600364304,

%U 3589048673,7896510620,17067607791,36283650153,75947406513,156672628539,318804641925,640390347979

%N Expansion of exp( Sum_{n>=1} sigma(6*n)*x^n/n ) in powers of x.

%C sigma(6*n) = A000203(6*n), the sum of divisors of 6*n (A224613).

%H Seiichi Manyama, <a href="/A283119/b283119.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{n>=1} (1 - x^(2*n))^4 * (1 - x^(3*n))^3/((1 - x^n)^12 * (1 - x^(6*n))).

%F a(n) = (1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - _Seiichi Manyama_, Mar 05 2017

%F a(n) ~ 55^(7/4) * exp(sqrt(55*n)*Pi/3) / (41472*sqrt(3)*n^(9/4)). - _Vaclav Kotesovec_, Mar 20 2017

%e G.f.: A(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 + ...

%e log(A(x)) = 12*x + 28*x^2/2 + 39*x^3/3 + 60*x^4/4 + 72*x^5/5 + 91*x^6/6 + 96*x^7/7 + 124*x^8/8 + ... + sigma(6*n)*x^n/n + ...

%t Table[SeriesCoefficient[Product[(1 - x^(2 i))^4*(1 - x^(3 i))^3/((1 - x^i)^12*(1 - x^(6 i))), {i, n}], {x, 0, n}], {n, 0, 25}] (* _Michael De Vlieger_, Mar 01 2017 *)

%Y Cf. A224613 (sigma(6*n)), A283164 (exp( Sum_{n>=1} -sigma(6*n)*x^n/n )).

%Y Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), this sequence (k=6), A283077 (k=7), A283120 (k=8), A283121 (k=9).

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 01 2017