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A338679 Expansion of Product_{k>=1} 1 / (1 - 10^(k-1)*x^k). 8
1, 1, 11, 111, 1211, 12211, 133211, 1343211, 14553211, 147653211, 1589753211, 16120753211, 173641753211, 1759951753211, 18855161753211, 192028261753211, 2048080361753211, 20841811361753211, 222333332361753211, 2261780642361753211, 24033895852361753211, 245331468952361753211 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
In general, if g.f. = Product_{k>=1} 1/(1 - d^(k-1)*x^k), where d > 1, then a(n) ~ sqrt(d-1) * polylog(2, 1/d)^(1/4) * d^(n - 1/2) * exp(2*sqrt(polylog(2, 1/d)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021
LINKS
FORMULA
a(n) = Sum_{k=0..n} p(n,k) * 10^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ 3 * polylog(2, 1/10)^(1/4) *10^(n - 1/2) * exp(2*sqrt(polylog(2, 1/10)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021
MATHEMATICA
nmax = 21; CoefficientList[Series[Product[1/(1 - 10^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 10^(n - k), {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 10^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]
CROSSREFS
Sequence in context: A015456 A343355 A240367 * A199764 A097177 A136982
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 23 2021
STATUS
approved

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Last modified July 17 13:32 EDT 2024. Contains 374377 sequences. (Running on oeis4.)