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%I #13 May 09 2021 02:30:07
%S 1,1,5,21,101,421,2021,8421,39397,167397,766437,3244517,14881253,
%T 62804453,283415013,1210159589,5401907685,22966866405,102497423845,
%U 435085808101,1925197238757,8215432696293,36068400468453,153579729097189,674546796630501,2866238341681637,12508012102193637
%N Expansion of Product_{k>=1} 1 / (1 - 4^(k-1)*x^k).
%F a(n) = Sum_{k=0..n} p(n,k) * 4^(n-k), where p(n,k) = number of partitions of n into k parts.
%F a(n) ~ sqrt(3) * polylog(2, 1/4)^(1/4) * 4^(n - 1/2) * exp(2*sqrt(polylog(2, 1/4)*n)) / (2*sqrt(Pi)*n^(3/4)). - _Vaclav Kotesovec_, May 09 2021
%t nmax = 26; CoefficientList[Series[Product[1/(1 - 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t Table[Sum[Length[IntegerPartitions[n, {k}]] 4^(n - k), {k, 0, n}], {n, 0, 26}]
%t a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]
%Y Cf. A008284, A075900, A246936, A300579, A338674, A338675, A338676, A338677, A338678, A338679.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 23 2021
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