0,3

This is the total number of hierarchies of n labeled elements arranged on 1 to n levels. A distribution of elements onto levels is "hierarchical" if a level l+1 contains <= elements than level l. Thus for n=4 the arrangement {1,2}:{3}{4} is not allowed. See also A140585. Examples: Let the colon ":" separate two consecutive levels l and l+1. Then n=2 --> 3: {1}{2}, {1}:{2}, {2}:{1}, n=3 --> 10: {1}{2}{3}, {1}{2}:{3}, {3}{1}:{2}, {2}{3}:{1}, {1}:{2}:{3}, {3}:{1}:{2}, {2}:{3}:{1}, {1}:{3}:{2}, {2}:{1}:{3}, {3}:{2}:{1}. - Thomas Wieder (thomas.wieder(AT)t-online.de), May 17 2008

n identical objects are painted by dipping them into a long row of cans of paint of distinct colors. Begining with the first can and not skipping any cans k, 1<=k<=n, objects are dipped (painted) and not more objects are dipped into any subsequent can than were dipped into the previous can. The painted objects are then linearly ordered. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jun 08 2009]

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

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126.

T. D. Noe, Table of n, a(n) for n=0..100

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

E.g.f.: 1 / Product (1 - x^k/k!).

a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(-k/d). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 14 2002

For n=3, say the first three cans in the row contain red, white, and blue paint respectively. The objects can be painted r,r,r or r,r,w or r,w,b and then linearly ordered in 1 + 3 + 6 = 10 ways. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jun 08 2009]

Table[Total[n!/Map[Function[n, Apply[Times, n! ]], Partitions[n]]], {n, 0, 20}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jun 08 2009]

Cf. A036038, A007837.

Cf. A140585.

Sequence in context: A020008 A000849 A092429 this_sequence A105748 A140964 A005921

Adjacent sequences: A005648 A005649 A005650 this_sequence A005652 A005653 A005654

nonn,easy,nice

Simon Plouffe (simon.plouffe(AT)gmail.com)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003