This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A099262 #43 Jan 05 2025 23:41:15
%S A099262 1,2,5,15,52,203,877,4139,21110,115179,665479,4030523,25343488,
%T A099262 164029595,1084948961,7291973067,49582466986,339971207051,
%U A099262 2345048898523,16244652278171,112871151708404,785938550025147,5480960778389365,38264428799608235,267342497477336542,1868866831126685483
%N 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)).
%C A099262 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.
%H A099262 Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%H A099262 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 A099262 N. Moreira and R. Reis, <a href="http://www.dcc.fc.up.pt/dcc/Pubs/TReports/TR04/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 A099262 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (22,-190,820,-1849,2038,-840).
%F A099262 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.
%F A099262 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
%F A099262 a(n) = Sum_{k=0..7} Stirling2(n,k).
%F A099262 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
%t A099262 Table[Sum[StirlingS2[n, k], {k, 0, 7}], {n, 1, 30}] (* _Robert A. Russell_, Apr 25 2018 *)
%o A099262 (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
%Y A099262 Cf. A007051, A007581, A056272, A056273, A099263.
%Y A099262 A row of the array in A278984.
%K A099262 easy,nonn
%O A099262 1,2
%A A099262 _Nelma Moreira_, Oct 10 2004
%E A099262 More terms from _Michel Marcus_, Jan 05 2025