A370765 - OEIS

%I #9 Mar 09 2024 03:48:39

%S 1,15,153,11295,31968,5289300,41957514,3216919050,-21009764691,

%T 2153132775315,-16978376482767,1659596014366335,-35929151338082922,

%U 1473739361689662990,-38968782475183427016,1541715187631618436300,-46858796372722560413526,1615119529247884664988030

%N a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k))^(1/3).

%F G.f.: Product_{k>=1} ((1 + 2^(k+1)*(9*x)^k) * (1 + 2^(k-1)*(9*x)^k))^(1/3).

%F a(n) ~ (-1)^(n+1) * c * 36^n / n^(4/3), where c = 0.244280405759762854740979712556383125782589356973734984...

%t nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]

%t nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*(9*x)^k)*(1+2^(k-1)*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]

%t nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, 2*x]*QPochhammer[-1/2, 2*x]/9)^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]

%t nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x]/9)^(1/3), {x, 0, nmax}], x] * 18^Range[0, nmax]

%Y Cf. A032302, A304961, A370016, A370761, A370764.

%K sign

%O 0,2

%A _Vaclav Kotesovec_, Mar 01 2024