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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)).
8
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
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
A row of the array in A278984.
Sequence in context: A287278 A287256 A287668 * A141081 A108305 A229224
KEYWORD
easy,nonn
AUTHOR
Nelma Moreira, Oct 10 2004
STATUS
approved