A338679 - OEIS

%I #11 May 09 2021 02:34:01

%S 1,1,11,111,1211,12211,133211,1343211,14553211,147653211,1589753211,

%T 16120753211,173641753211,1759951753211,18855161753211,

%U 192028261753211,2048080361753211,20841811361753211,222333332361753211,2261780642361753211,24033895852361753211,245331468952361753211

%N Expansion of Product_{k>=1} 1 / (1 - 10^(k-1)*x^k).

%C 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

%F a(n) = Sum_{k=0..n} p(n,k) * 10^(n-k), where p(n,k) = number of partitions of n into k parts.

%F 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

%t nmax = 21; CoefficientList[Series[Product[1/(1 - 10^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t Table[Sum[Length[IntegerPartitions[n, {k}]] 10^(n - k), {k, 0, n}], {n, 0, 21}]

%t 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}]

%Y Cf. A008284, A075900, A246942, A300579, A338673, A338674, A338675, A338676, A338677, A338678.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 23 2021