A002874 The number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles. (Formerly M1863 N0738)

1, 2, 8, 42, 268, 1994, 16852, 158778, 1644732, 18532810, 225256740, 2933174842, 40687193548, 598352302474, 9290859275060, 151779798262202, 2600663778494172, 46609915810749130, 871645673599372868, 16971639450858467002, 343382806080459389676

Offset

0,2

Comments

Original name: Sorting numbers.

References

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

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..484 (first 101 terms from T. D. Noe)

Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.

Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.

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]

J. Pasukonis, S. Ramgoolam, From counting to construction for BPS states in N=4SYM, arXiv:1010.1683 [hep-th], 2010, (E.3).

J. Pasukonis, S. Ramgoolam, From counting to construction for BPS states in N=4SYM, J. High En. Phys. 2011 (2) (2011), (E.3).

OEIS Wiki, Sorting numbers

Index entries for sequences related to sorting

Formula

E.g.f.: exp( (exp(3*x) - 4)/3 + exp(x) ).

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

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

Maple

S:= series(exp( (exp(3*x) - 4)/3 + exp(x)), x, 31):
seq(coeff(S, x, j)*j!, j=0..30); # Robert Israel, Oct 30 2015
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((1+
      3^(j-1))*binomial(n-1, j-1)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 17 2017

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, 3], {n, 0, 12}] (* Wouter Meeussen, Dec 06 2008 *)
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 *)
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 *)

Crossrefs

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

Equals column 3 of A162663. - Michel Marcus, Mar 27 2013

Row sums of A294201.

Sequence in context: A005315 A182520 A121635 * A324961 A351814 A078592

Adjacent sequences: A002871 A002872 A002873 * A002875 A002876 A002877

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Simon Plouffe

Extensions

New name from Danny Rorabaugh, Oct 24 2015

Status

approved