A257863 Expansion of 1/(1 - x - x^2 + x^5 - x^6).

1, 1, 2, 3, 5, 7, 12, 18, 29, 45, 72, 112, 178, 279, 441, 693, 1094, 1721, 2714, 4273, 6735, 10607, 16715, 26329, 41485, 65352, 102965, 162209, 255560, 402613, 634306, 999306, 1574368, 2480323, 3907638, 6156268, 9698906, 15280112, 24073063, 37925860, 59750293

Offset

0,3

Comments

This sequence counts partially ordered partitions of (n) into parts (1,2,3,4) where only the position (order) of the 4's are important. The 4's behave like placeholders for the unordered 1's, 2's and 3's. (See example.)

Links

Table of n, a(n) for n=0..40.

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

Formula

G.f.: 1/(1-x-x^2+x^5-x^6).

a(n) = a(n-1) + a(n-2) - a(n-5) + a(n-6).

Example

a(8)=29 These are (44),(341),(143),(431=413),(314=134),(422),(242),(224),(4211=4121=4112),(2114=1214=1124),(1421=1412),(2141=1241),(2411),(1142),(41111),(14111),(11411),(11141),(11114),(332=323=233),(3311=1133=1331=3113=1313=3131),(3221=twelve),(32111=twenty),(311111=six),(2222),(22211=ten),(221111=fifteen),(2111111=seven),(11111111)

Mathematica

RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] - a[n - 5] + a[n - 6], a[1] == 1, a[2] == 1, a[3] == 2, a[4] == 3, a[5] == 5, a[6] == 7}, a, {n, 43}] (* Michael De Vlieger, May 11 2015 *)
CoefficientList[Series[1/(1 - x - x^2 + x^5 - x^6), {x, 0, 80}], x] (* or *) LinearRecurrence[{1, 1, 0, 0, -1, 1}, {1, 1, 2, 3, 5, 7}, 50] (* Vincenzo Librandi, May 12 2015 *)

Prog

(Magma) [n le 6 select NumberOfPartitions(n-1) else Self(n-1)+Self(n-2)-Self(n-5)+Self(n-6): n in [1..50]]; // Vincenzo Librandi, May 12 2015
(Sage) m = 50; L.<x> = PowerSeriesRing(ZZ, m); f = 1/(1-x-x^2+x^5-x^6); print(f.coefficients()) # Bruno Berselli, May 12 2015

Crossrefs

Sequence in context: A263358 A239915 A013983 * A169986 A218021 A137713

Adjacent sequences: A257860 A257861 A257862 * A257864 A257865 A257866

Keyword

nonn,easy

Author

David Neil McGrath, May 11 2015

Status

approved