|
| |
|
|
A000478
|
|
Number of ways of placing n labeled balls into 3 indistinguishable boxes with at least 2 balls in each box.
(Formerly M4978 N2138)
|
|
4
| |
|
|
15, 105, 490, 1918, 6825, 22935, 74316, 235092, 731731, 2252341, 6879678, 20900922, 63259533, 190957923, 575363776, 1731333808, 5205011031, 15638101281, 46962537810, 140988276150, 423174543025, 1269959836015
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 6,1
|
|
|
COMMENTS
| Associated Stirling numbers.
|
|
|
REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
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
| T. D. Noe, Table of n, a(n) for n=6..200
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
|
|
|
FORMULA
| E.g.f.: ((exp(x) - 1 - x)^3)/3!.
G.f.: x^6*(12*x^3 - 40*x^2 + 45*x - 15)/((1 - x)^3*(1 - 2*x)^2*(3*x - 1))
a(n) = (1+n+n^2)/2 - (1/2 + n/4)*2^n + 3^n/6 - Michael Steyer (m.steyer(AT)osram.de), Jan 09 2005
a(6)=15, a(7)=105, a(8)=490, a(9)=1918, a(10)=6825, a(11)=22935, a(n)=10*a(n-1)-40*a(n-2)+82*a(n-3)-91*a(n-4)+52*a(n-5)-12*a(n-6) [From Harvey P. Dale based on Michael Steyer's formula, Jul 17 2011]
|
|
|
EXAMPLE
| a(6)=6!/(2!*2!*2!*3!)=15
|
|
|
MAPLE
| A000478:=-(-15+45*z-40*z**2+12*z**3)/(-1+3*z)/(2*z-1)**2/(z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation.]
|
|
|
MATHEMATICA
| Table[(1+n+n^2)/2-(1/2+n/4)*2^n+3^n/6, {n, 6, 30}] (* or *) LinearRecurrence[ {10, -40, 82, -91, 52, -12}, {15, 105, 490, 1918, 6825, 22935}, 25] (* From Harvey P. Dale, Jul 17 2011 *)
|
|
|
CROSSREFS
| Cf. A000247 (2 boxes), A058844 (4 boxes).
Sequence in context: A076767 A022610 A006857 * A055848 A202493 A200852
Adjacent sequences: A000475 A000476 A000477 * A000479 A000480 A000481
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Additional comments from Michael Steyer (msteyer(AT)osram.de), Dec 02 2000. More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 06 2000
|
| |
|
|
