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%I #54 Sep 08 2022 08:45:15
%S 1,2,5,15,52,203,877,4140,21146,115929,677359,4189550,27243100,
%T 184941915,1301576801,9433737120,69998462014,529007272061,
%U 4054799902003,31415584940850,245382167055488,1928337630016767,15222915798289765
%N 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)).
%C 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 a regular expression with c = 8: Sum_{i=1..c} (Product_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation.
%H Vincenzo Librandi, <a href="/A099263/b099263.txt">Table of n, a(n) for n = 1..1000</a>
%H Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%H N. Moreira and R. Reis, <a href="http://www.dcc.fc.up.pt/~nam/publica/dcc-2004-07.pdf">On the density of languages representing finite set partitions</a>, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
%H N. Moreira and R. Reis, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Moreira/moreira8.html">On the Density of Languages Representing Finite Set Partitions</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (29,-343,2135,-7504,14756,-14832,5760).
%F 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)! for c > 1, and g(k, c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.
%F 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]
%F a(n) = Sum_{k=0..8} Stirling2(n,k).
%F G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} (1 - j*x) with k = 8. - _Robert A. Russell_, Apr 25 2018
%t 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 *)
%t Table[Sum[StirlingS2[n, k], {k, 0, 8}], {n, 1, 30}] (* _Robert A. Russell_, Apr 25 2018 *)
%t LinearRecurrence[{29,-343,2135,-7504,14756,-14832,5760},{1,2,5,15,52,203,877},30] (* _Harvey P. Dale_, Aug 27 2019 *)
%o (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
%o (PARI) 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; \\ _Altug Alkan_, Apr 25 2018
%Y Cf. A007051, A007581, A056272, A056273, A099262.
%Y A row of the array in A278984.
%Y Cf. A008277 (Stirling2), A248925.
%K nonn,easy
%O 1,2
%A _Nelma Moreira_, Oct 10 2004
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