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A338677
Expansion of Product_{k>=1} 1 / (1 - 8^(k-1)*x^k).
9
1, 1, 9, 73, 649, 5257, 46729, 378505, 3331721, 27219593, 237491849, 1938544265, 16925054601, 138041874057, 1196384310921, 9820024329865, 84609648809609, 693596417152649, 5977550934234761, 48976660041553545, 419984680697190025, 3455551232025810569, 29494747047731910281
OFFSET
0,3
FORMULA
a(n) = Sum_{k=0..n} p(n,k) * 8^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ sqrt(7) * polylog(2, 1/8)^(1/4) * 8^(n - 1/2) * exp(2*sqrt(polylog(2, 1/8)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021
MATHEMATICA
nmax = 22; CoefficientList[Series[Product[1/(1 - 8^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 8^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 8^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 23 2021
STATUS
approved