%I #5 Jun 11 2015 06:23:49

%S 1,1,3,13,61,324,1800,10340,60969,366486,2237120,13829487,86394782,

%T 544547651,3458637273,22113504345,142212705879,919294844898,

%U 5969839457411,38927450022860,254776529381625,1673102335692514,11020847332241873,72798664086854460

%N a(n) = [x^n] Product_{k=1..n} (1+x^k)^3 / x^(2*k).

%F a(n) ~ c * d^n / n^(3/2), where d = 7.036711302278424796297167109247361745558645910729132828752853658917..., c = 0.282321145891... .

%t Table[SeriesCoefficient[Product[(1+x^k)^3/x^(2*k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]

%t Table[SeriesCoefficient[Product[1+x^k, {k, 1, n}]^3, {x, 0, n*(n+2)}], {n, 0, 30}]

%t (* A program to compute the constant d *) (1+r)^3/r /.FindRoot[Log[1+r]/Log[r] + (PolyLog[2,-r] + Pi^2/12) / Log[r]^2 == 1/6, {r, E}, WorkingPrecision->100]

%Y Cf. A258797, A258798.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jun 10 2015