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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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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).
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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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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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