A370712 - OEIS

%I #8 Feb 28 2024 18:08:07

%S 1,3,0,99,-270,2430,-10287,105462,-750141,5702481,-42623901,347424633,

%T -2779077762,22353287634,-181730796723,1493711042589,-12321529794261,

%U 102125312638713,-850797139405887,7120067746384863,-59800770201017934,503922807927384129,-4259721779079782751

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

%F G.f.: Product_{k>=1} (1 + 3*(3*x)^k)^(1/3).

%F a(n) ~ (-1)^(n+1) * c * 9^n / n^(4/3), where c = QPochhammer(-1/3)^(1/3) / (3*Gamma(2/3)) = 0.26286302373105271371291957730496322329245126572...

%t nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]

%t nmax = 30; CoefficientList[Series[Product[(1 + 3*(3*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]

%t nmax = 30; CoefficientList[Series[(QPochhammer[-3, x]/4)^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]

%Y Cf. A032308, A370710.

%Y Cf. A300579, A344062.

%K sign

%O 0,2

%A _Vaclav Kotesovec_, Feb 27 2024