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A099265
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a(5,n) := 1/96 5^n + 1/8 3^n + + 1/3 2^n + 3/8 n - 15/32 Partial sum of A056272 (= sum_{m=1..n}sum_{i=1..5}S(m,i)) (i.e. partial sum of Stirling numbers of second kind S(n,i) for i=1..5)).
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1
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1, 3, 8, 23, 75, 277, 1132, 4977, 22979, 109451, 531456, 2610931, 12917683, 64181625, 319695980, 1594859885, 7963472187, 39784944799, 198827606704, 993846943839, 4968361974491, 24839192686973, 124188113975628
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Density of regular language L{0}* over {0,1,2,3,4,5} (i.e. number of strings of length n), where L is described by regular expression with c=5: sum_{i=1..c}(prod_{j=1..i}(j(1+...+j)*) where sum stands for union and prod for concatenation. I.e L=L((11*+11*2(1+2)*+..+11*2(1+2)*3(1+2+3)*4(1+2+3+4)*5(1+2+3+4+5)*)0*)
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REFERENCES
| 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
| dcc-2004-07.ps
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FORMULA
| For c=5, a(c, n)=g(1, c)*n+sum_{k=2..c}((g(k, c)*k*(k^n - 1))/(k - 1)) 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*(-1+19*x^3-24*x^2+9*x)/((3*x-1)*(2*x-1)*(5*x-1)*(x-1)^2) [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]
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MAPLE
| with (combinat):seq(sum(sum(stirling2(k, j), j=1..5), k=1..n), n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 04 2007
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CROSSREFS
| Cf. A056272, A047926, A099266, A099264.
Sequence in context: A050511 A151405 A148778 * A099266 A024716 A189359
Adjacent sequences: A099262 A099263 A099264 * A099266 A099267 A099268
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KEYWORD
| easy,nonn
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AUTHOR
| Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004
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