0,3

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.

Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,1,0,0,-1,1).

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 (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).

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 *)

(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

(SageMath) 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

Sequence in context: A097596 A054191 A347761 * A079975 A253511 A076739

Adjacent sequences: A257789 A257790 A257791 * A257793 A257794 A257795

nonn,easy

David Neil McGrath, May 08 2015

More terms from Vincenzo Librandi, May 09 2015

approved