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 A257792 Expansion of 1/(1-x-x^2-x^3-x^5+x^8-x^9). 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,1,0,0,-1,1). FORMULA 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). EXAMPLE 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). MATHEMATICA 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 *) PROG (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. = 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 CROSSREFS Sequence in context: A097596 A054191 A347761 * A079975 A253511 A076739 Adjacent sequences: A257789 A257790 A257791 * A257793 A257794 A257795 KEYWORD nonn,easy AUTHOR David Neil McGrath, May 08 2015 EXTENSIONS More terms from Vincenzo Librandi, May 09 2015 STATUS approved

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Last modified November 29 04:45 EST 2022. Contains 358422 sequences. (Running on oeis4.)