login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A099263 a(n) = (1/40320) 8^n + (1/1440) 6^n + (1/360) 5^n + (1/64) 4^n + (11/180) 3^n + (53/288) 2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)). 8
1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115929, 677359, 4189550, 27243100, 184941915, 1301576801, 9433737120, 69998462014, 529007272061, 4054799902003, 31415584940850, 245382167055488, 1928337630016767, 15222915798289765 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Density of regular language L over {1,2,3,4,5,6,7,8} (i.e., number of strings of length n in L) described by regular expression with c=8: sum_{i=1..c} (prod_{j=1..i}(j(1+..+j)*) where sum stands for union and prod for concatenation.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

N. Moreira and R. Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.

N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.

Index entries for linear recurrences with constant coefficients, signature (29,-343,2135,-7504,14756,-14832,5760).

FORMULA

For c=8, a(n) = (c^n)/c! + Sum_{k=1..c-2} ((k^n)/k!*(Sum_{j=2..c-k} (((-1)^j)/j!))) or = Sum_{k=1..c} (g(k, c)*k^n) where g(1, 1) = 1 g(1, c) = g(1, c-1) + ((-1)^(c-1))/(c-1)!, c > 1 g(k, c) = g(k-1, c-1)/k, for c > 1 and 2 <= k <= c.

G.f.: -x*(3641*x^6 - 6583*x^5 + 4566*x^4 - 1579*x^3 + 290*x^2 - 27*x + 1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). [Colin Barker, Dec 05 2012]

MATHEMATICA

CoefficientList[Series[-(3641 x^6 - 6583 x^5 + 4566 x^4 - 1579 x^3 + 290 x^2 - 27 x + 1) / ((x - 1) (2 x - 1) (3 x - 1) (4 x - 1) (5 x - 1) (6 x - 1) (8 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 27 2017 *)

PROG

(MAGMA) [(1/40320)*8^n+(1/1440)*6^n+(1/360)*5^n+(1/64)*4^n +(11/180)*3^n+(53/288)*2^n+103/280: n in [1..30]]; // Vincenzo Librandi, Jul 27 2017

CROSSREFS

Cf. A007051, A007581, A056272, A056273, A099262.

A row of the array in A278984.

Sequence in context: A287279 A287257 A287669 * A192865 A229225 A276725

Adjacent sequences:  A099260 A099261 A099262 * A099264 A099265 A099266

KEYWORD

easy,nonn

AUTHOR

Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy .

Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.