login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002874 The number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles.
(Formerly M1863 N0738)
17

%I M1863 N0738 #80 Jul 22 2022 02:24:46

%S 1,2,8,42,268,1994,16852,158778,1644732,18532810,225256740,2933174842,

%T 40687193548,598352302474,9290859275060,151779798262202,

%U 2600663778494172,46609915810749130,871645673599372868,16971639450858467002,343382806080459389676

%N The number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles.

%C Original name: Sorting numbers.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A002874/b002874.txt">Table of n, a(n) for n = 0..484</a> (first 101 terms from T. D. Noe)

%H Vaclav Kotesovec, <a href="/A002874/a002874.jpg">Graph - the asymptotic ratio (10000 terms)</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 Victor Meally, <a href="/A002868/a002868.pdf">Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.</a>

%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 J. Pasukonis, S. Ramgoolam, <a href="https://arxiv.org/abs/1010.1683">From counting to construction for BPS states in N=4SYM</a>, arXiv:1010.1683 [hep-th], 2010, (E.3).

%H J. Pasukonis, S. Ramgoolam, <a href="https://doi.org/10.1007/JHEP02(2011)078">From counting to construction for BPS states in N=4SYM</a>, J. High En. Phys. 2011 (2) (2011), (E.3).

%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(3*x) - 4)/3 + exp(x) ).

%F a(n) ~ exp(exp(3*r)/3 + exp(r) - 4/3 - n) * (n/r)^(n + 1/2) / sqrt((1 + 3*r)*exp(3*r) + (1 + r)*exp(r)), where r = LambertW(3*n)/3 - 1/(1 + 3/LambertW(3*n) + n^(2/3) * (1 + LambertW(3*n)) * (3/LambertW(3*n))^(5/3)). - _Vaclav Kotesovec_, Jul 03 2022

%F a(n) ~ (3*n/LambertW(3*n))^n * exp(n/LambertW(3*n) + (3*n/LambertW(3*n))^(1/3) - n - 4/3) / sqrt(1 + LambertW(3*n)). - _Vaclav Kotesovec_, Jul 10 2022

%p S:= series(exp( (exp(3*x) - 4)/3 + exp(x)), x, 31):

%p seq(coeff(S,x,j)*j!, j=0..30); # _Robert Israel_, Oct 30 2015

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=0, 1, add((1+

%p 3^(j-1))*binomial(n-1, j-1)*a(n-j), j=1..n))

%p end:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 17 2017

%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,3],{n,0,12}] (* _Wouter Meeussen_, Dec 06 2008 *)

%t mx = 16; p = 3; 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] * 3^k * BellB[k, 1/3] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)

%Y u[n,j] generates for j=1, A000110; j=2, A002872; j=3, this sequence; j=4, A141003; j=5, A036075; j=6, A141004; j=7, A036077. - _Wouter Meeussen_, Dec 06 2008

%Y Equals column 3 of A162663. - _Michel Marcus_, Mar 27 2013

%Y Row sums of A294201.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_

%E New name from _Danny Rorabaugh_, Oct 24 2015

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)