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A257792
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Expansion of 1/(1-x-x^2-x^3-x^5+x^8-x^9).
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0
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1, 1, 2, 4, 7, 14, 26, 49, 92, 174, 328, 618, 1166, 2197, 4143, 7811, 14726, 27764, 52344, 98687, 186058, 350784, 661347, 1246865, 2350768, 4432000, 8355837, 15753609, 29700940, 55996428, 105572414, 199040101, 375258649, 707490872, 1333862213, 2514786376
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OFFSET
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0,3
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COMMENTS
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This sequence counts partially ordered partitions of (n) in two distinct ways. It partitions (n) into parts containing (1,2,3,5,9) where the adjacent order of 3's and 5's are unimportant, example (1), and it partitions (n) into parts containing (1,2,3,4,5,6) where the adjacent order of the odd numbers is unimportant, example (2). The sign "=" is used within a bracket to indicate that the arrangements are counted as one.
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LINKS
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FORMULA
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G.f.: 1/(1-x-x^2-x^3-x^5+x^8-x^9).
a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-5) - a(n-8) + a(n-9).
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EXAMPLE
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Example (1):Partial order of (n) into parts (1,2,3,5,9) where the adjacent order of 3's and 5's is unimportant. a(8)=92 These are (53=35)=1,(521)=6,(5111)=4,(332)=3,(3311)=6,(3221)=12,(32111)=20,(311111)=6,(2222)=1,(22211)=10,(221111)=15,(2111111)=7,(11111111)=1.
Example (2):Partial order of (n) into parts (1,2,3,4,5,6) where the adjacent order of all odd numbers (i.e. 1,3,5) is unimportant. a(6)=26 These are (6),(51=15),(42),(24),(411),(141),(114),(33),(321),(123),(231=213),(312=132),(3111=1311=1131=1113),(222),(2211),(1122),(1221),(2112),(2121),(1212),(21111),(12111),(11211),(11121),(11112),(111111).
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MATHEMATICA
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CoefficientList[Series[1/(1 - x - x^2 - x^3 - x^5 + x^8 - x^9), {x, 0, 80}], x] (* Vincenzo Librandi, May 09 2015 *)
LinearRecurrence[{1, 1, 1, 0, 1, 0, 0, -1, 1}, {1, 1, 2, 4, 7, 14, 26, 49, 92}, 36] (* Ray Chandler, Jul 14 2015 *)
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PROG
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(Magma) I:=[1, 1, 2, 4, 7, 14, 26, 49, 92]; [n le 9 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)+Self(n-5)-Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, May 09 2015
(Sage) m = 40; L.<x> = PowerSeriesRing(ZZ, m); f = 1/(1-x-x^2-x^3-x^5+x^8-x^9); print(f.coefficients()) # Bruno Berselli, May 12 2015
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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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EXTENSIONS
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STATUS
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approved
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