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, 112871151708404, 785938550025147, 5480960778389365, 38264428799608235, 267342497477336542, 1868866831126685483
Offset: 1
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).
Programs
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Mathematica
Table[Sum[StirlingS2[n, k], {k, 0, 7}], {n, 1, 30}] (* Robert A. Russell, Apr 25 2018 *) -
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
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
Extensions
More terms from Michel Marcus, Jan 05 2025
Comments