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%I #13 May 09 2021 02:28:16
%S 1,1,8,57,449,3193,25145,178809,1391314,9996498,76955586,552257546,
%T 4255024523,30502987019,232969386483,1682476714724,12762937304013,
%U 92019035596293,698222541789109,5030814634614406,37955614705675479,274741644961416648,2061916926761604144,14909943849253537057
%N Expansion of Product_{k>=1} 1 / (1 - 7^(k-1)*x^k).
%F a(n) = Sum_{k=0..n} p(n,k) * 7^(n-k), where p(n,k) = number of partitions of n into k parts.
%F a(n) ~ sqrt(6) * polylog(2, 1/7)^(1/4) * 7^(n - 1/2) * exp(2*sqrt(polylog(2, 1/7)*n)) / (2*sqrt(Pi)*n^(3/4)). - _Vaclav Kotesovec_, May 09 2021
%t nmax = 23; CoefficientList[Series[Product[1/(1 - 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t Table[Sum[Length[IntegerPartitions[n, {k}]] 7^(n - k), {k, 0, n}], {n, 0, 23}]
%t a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 7^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]
%Y Cf. A008284, A075900, A246939, A300579, A338673, A338674, A338675, A338677, A338678, A338679.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 23 2021
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