A344063 - OEIS

%I #9 May 09 2021 03:32:29

%S 1,1,4,20,80,384,1600,7424,30720,143360,593920,2703360,11403264,

%T 51118080,214958080,965738496,4047503360,17951621120,76168560640,

%U 334202142720,1411970498560,6211596451840,26203595472896,114246130073600,484815908372480,2101441598586880,8896148580335616

%N Expansion of Product_{k>=1} (1 + 4^(k-1)*x^k).

%F a(n) = Sum_{k=0..A003056(n)} q(n,k) * 4^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

%F a(n) ~ (-polylog(2, -1/4))^(1/4) * 4^n * exp(2*sqrt(-polylog(2, -1/4)*n)) / (2*sqrt(5*Pi/4)*n^(3/4)). - _Vaclav Kotesovec_, May 09 2021

%t nmax = 26; CoefficientList[Series[Product[(1 + 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 4^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 26}]

%o (PARI) seq(n)={Vec(prod(k=1, n, 1 + 4^(k-1)*x^k + O(x*x^n)))} \\ _Andrew Howroyd_, May 08 2021

%Y Cf. A003056, A008289, A261568, A304961, A338673, A340103, A344062, A344064, A344065, A344066, A344067, A344068.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 08 2021