%I #44 Jul 22 2022 02:20:24
%S 1,2,12,106,1144,14434,209736,3451290,63194936,1269555762,27700698344,
%T 651497885482,16414347638936,440651469115394,12546081858835528,
%U 377328994871025210,11946046637611280120
%N The number of partitions of {1..7n} that are invariant under a permutation consisting of n 7-cycles.
%C Original name: Sorting numbers.
%H Vincenzo Librandi, <a href="/A036077/b036077.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=7.
%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=7. - _Vaclav Kotesovec_, Jul 03 2022
%F a(n) ~ (7*n/LambertW(7*n))^n * exp(n/LambertW(7*n) + (7*n/LambertW(7*n))^(1/7) - n - 8/7) / sqrt(1 + LambertW(7*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,7],{n,0,12}] (* _Wouter Meeussen_, Dec 06 2008 *)
%t mx = 16; p = 7; 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] * 7^k * BellB[k, 1/7] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)
%Y Cf. A001861, A002872-A002875, A036074.
%Y u[n,j] generates for j=1, A000110; j=2, A002872; j=3, A002874; j=4, A141003; j=5, A036075; j=6, A141004; j=7, this sequence. - _Wouter Meeussen_, Dec 06 2008
%Y Column 7 of A162663.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E New name from _Danny Rorabaugh_, Oct 24 2015