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%I #39 Jul 22 2022 16:42:38
%S 1,2,16,202,3044,52794,1055260,24081754,615896308,17347970202,
%T 531721375308,17595339114554,624882463734756,23691503493287738,
%U 954301756159098172,40665568780962213530,1826521141853468785364
%N The number of partitions of {1..(11n)} that are invariant under a permutation consisting of n 11-cycles.
%C Original name: Sorting numbers.
%H Vincenzo Librandi, <a href="/A036081/b036081.txt">Table of n, a(n) for n = 0..200</a>
%H Vaclav Kotesovec, <a href="https://arxiv.org/abs/2207.10568">Asymptotics for a certain group of exponential generating functions</a>, arXiv:2207.10568 [math.CO], Jul 13 2022.
%H T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
%H OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%H <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=11.
%F a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=11. - _Vaclav Kotesovec_, Jul 03 2022
%F a(n) ~ (11*n/LambertW(11*n))^n * exp(n/LambertW(11*n) + (11*n/LambertW(11*n))^(1/11) - n - 12/11) / sqrt(1 + LambertW(11*n)). - _Vaclav Kotesovec_, Jul 10 2022
%t u[0, j_] := 1; u[k_, j_] := u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 11], {n, 0, 30}] (* _Vincenzo Librandi_, Dec 12 2012 - after _Wouter Meeussen_ in similar sequences *)
%t mx = 16; p = 11; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* _Robert G. Wilson v_, Dec 12 2012 *)
%t Table[Sum[Binomial[n,k] * 11^k * BellB[k, 1/11] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)
%Y Cf. A001861, A002872, A002873, A002874, A002875, A036074.
%Y Column 11 of A162663.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E New name from _Danny Rorabaugh_, Oct 24 2015
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