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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

Table of n, a(n) for n=0..35.

Index entries for related partition-counting sequences

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.<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

CROSSREFS

Sequence in context: A088813 A097596 A054191 * 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 May 27 21:12 EDT 2020. Contains 334671 sequences. (Running on oeis4.)