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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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8
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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 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.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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, 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.
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)! for c > 1, and 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]
a(n) = Sum_{k=0..8} Stirling2(n,k).
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
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MATHEMATICA
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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 *)
Table[Sum[StirlingS2[n, k], {k, 0, 8}], {n, 1, 30}] (* Robert A. Russell, Apr 25 2018 *)
LinearRecurrence[{29, -343, 2135, -7504, 14756, -14832, 5760}, {1, 2, 5, 15, 52, 203, 877}, 30] (* Harvey P. Dale, Aug 27 2019 *)
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PROG
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(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
(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
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CROSSREFS
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Cf. A007051, A007581, A056272, A056273, A099262.
A row of the array in A278984.
Cf. A008277 (Stirling2), A248925.
Sequence in context: A287279 A287257 A287669 * A192865 A229225 A343669
Adjacent sequences: A099260 A099261 A099262 * A099264 A099265 A099266
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KEYWORD
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nonn,easy
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AUTHOR
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Nelma Moreira, Oct 10 2004
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STATUS
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approved
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