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A099263
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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)).
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7
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1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115929, 677359, 4189550, 27243100, 184941915, 1301576801, 9433737120, 69998462014, 529007272061, 4054799902003, 31415584940850, 245382167055488, 1928337630016767, 15222915798289765
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OFFSET
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1,2
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COMMENTS
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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 regular expression with c=8: sum_{i=1..c}(prod_{j=1..i}(j(1+..+j)*) where sum stands for union and prod for concatenation.
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REFERENCES
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Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
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.
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LINKS
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Table of n, a(n) for n=1..23.
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
N. Moreira and R. Reis, dcc-2004-07.ps
Index entries for linear recurrences with constant coefficients, signature (29,-343,2135,-7504,14756,-14832,5760).
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FORMULA
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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)!, c>1 g(k, c)=g(k-1, c-1)/k, for c>1 and 2<= k<= c
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]
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CROSSREFS
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Cf. A007051, A007581, A056272, A056273, A099262.
A row of the array in A278984.
Sequence in context: A287279 A287257 A287669 * A192865 A229225 A276725
Adjacent sequences: A099260 A099261 A099262 * A099264 A099265 A099266
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KEYWORD
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easy,nonn
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AUTHOR
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Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004
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STATUS
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approved
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