A099262 a(n) = (1/5040)*7^n + (1/240)*5^n + (1/72)*4^n + (1/16)*3^n + (11/60)*2^n + 53/144. Partial sum of Stirling numbers of second kind S(n,i), i=1..7 (i.e., a(n) = Sum_{i=1..7} S(n,i)).

1, 2, 5, 15, 52, 203, 877, 4139, 21110, 115179, 665479, 4030523, 25343488, 164029595, 1084948961, 7291973067, 49582466986, 339971207051, 2345048898523, 16244652278171

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..20.

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, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.

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.

Index entries for linear recurrences with constant coefficients, signature (22,-190,820,-1849,2038,-840).

Formula

For c=7, 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*(531*x^5-881*x^4+535*x^3-151*x^2+20*x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)). - Colin Barker, Dec 05 2012

a(n) = Sum_{k=0..7} Stirling2(n,k).

G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=7. - Robert A. Russell, Apr 25 2018

Mathematica

Table[Sum[StirlingS2[n, k], {k, 0, 7}], {n, 1, 30}] (* Robert A. Russell, Apr 25 2018 *)

Prog

(PARI) a(n) = (1/5040)*7^n + (1/240)*5^n + (1/72)*4^n + (1/16)*3^n + (11/60)*2^n + 53/144; \\ Altug Alkan, Apr 25 2018

Crossrefs

Cf. A007051, A007581, A056272, A056273, A099263.

A row of the array in A278984.

Sequence in context: A287278 A287256 A287668 * A141081 A108305 A229224

Adjacent sequences: A099259 A099260 A099261 * A099263 A099264 A099265

Keyword

easy,nonn

Author

Nelma Moreira, Oct 10 2004

Status

approved