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
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.
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
New name from Danny Rorabaugh, Oct 24 2015
STATUS
approved