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%I #11 May 09 2021 02:31:15
%S 1,1,10,91,901,8191,81091,737191,7239142,66288142,646149322,
%T 5912729632,57664985653,527352541453,5111015223223,46998961540624,
%U 453182267869615,4163124744738505,40151590267580785,368699990679135946,3540322181970716707,32632895079429817528,312061810101214595698
%N Expansion of Product_{k>=1} 1 / (1 - 9^(k-1)*x^k).
%F a(n) = Sum_{k=0..n} p(n,k) * 9^(n-k), where p(n,k) = number of partitions of n into k parts.
%F a(n) ~ sqrt(2) * polylog(2, 1/9)^(1/4) * 9^(n - 1/2) * exp(2*sqrt(polylog(2, 1/9)*n)) / (sqrt(Pi)*n^(3/4)). - _Vaclav Kotesovec_, May 09 2021
%t nmax = 22; CoefficientList[Series[Product[1/(1 - 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t Table[Sum[Length[IntegerPartitions[n, {k}]] 9^(n - k), {k, 0, n}], {n, 0, 22}]
%t a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 9^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]
%Y Cf. A008284, A075900, A246941, A300579, A338673, A338674, A338675, A338676, A338677, A338679.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 23 2021
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