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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 (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} (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 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

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Last modified June 16 04:56 EDT 2021. Contains 345056 sequences. (Running on oeis4.)