A036077 The number of partitions of {1..7n} that are invariant under a permutation consisting of n 7-cycles.

1, 2, 12, 106, 1144, 14434, 209736, 3451290, 63194936, 1269555762, 27700698344, 651497885482, 16414347638936, 440651469115394, 12546081858835528, 377328994871025210, 11946046637611280120

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Original name: Sorting numbers.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
• T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
• OEIS Wiki, Sorting numbers
• Index entries for sequences related to sorting

Crossrefs

Cf. A001861, A002872-A002875, A036074.

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

Column 7 of A162663.

Programs

Mathematica

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 *)
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 *)
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 *)

Formula

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=7.

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

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

Extensions

New name from Danny Rorabaugh, Oct 24 2015