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A036075 The number of partitions of {1..5n} that are invariant under a permutation consisting of n 5-cycles.

%I A036075 #40 Jul 22 2022 02:22:36
%S A036075 1,2,10,70,602,6078,70402,917830,13253002,209350350,3584098770,
%T A036075 66012131222,1300004931162,27232369503902,604103160535330,
%U A036075 14136908333006822,347827448896896554,8971450949011952494
%N A036075 The number of partitions of {1..5n} that are invariant under a permutation consisting of n 5-cycles.
%C A036075 Original name: Sorting numbers.
%H A036075 Vincenzo Librandi, <a href="/A036075/b036075.txt">Table of n, a(n) for n = 0..200</a>
%H A036075 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 A036075 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 A036075 OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%H A036075 <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F A036075 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=5.
%F A036075 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=5. - _Vaclav Kotesovec_, Jul 03 2022
%F A036075 a(n) ~ (5*n/LambertW(5*n))^n * exp(n/LambertW(5*n) + (5*n/LambertW(5*n))^(1/5) - n - 6/5) / sqrt(1 + LambertW(5*n)). - _Vaclav Kotesovec_, Jul 10 2022
%t A036075 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,5],{n,0,12}] (* _Wouter Meeussen_, Dec 06 2008 *)
%t A036075 mx = 16; p = 5; 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 A036075 Table[Sum[Binomial[n,k] * 5^k * BellB[k, 1/5] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)
%Y A036075 Cf. A001861, A002872-A002875, A036074.
%Y A036075 u[n,j] generates for j=1, A000110 Bell numbers; j=2, A002872; j=3, A002874; j=4, A141003 (Mathar); j=5, this sequence; j=6, A141004 (Mathar); j=7, A036077. - _Wouter Meeussen_, Dec 06 2008
%Y A036075 Column 5 of A162663.
%K A036075 nonn
%O A036075 0,2
%A A036075 _N. J. A. Sloane_
%E A036075 New name from _Danny Rorabaugh_, Oct 24 2015