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

Author

David Neil McGrath, May 11 2015

Keywords

• nonn
• easy

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.)

Examples

			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)
		

Links

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

Programs

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

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 *)

Sage

m = 50; L. = PowerSeriesRing(ZZ, m); f = 1/(1-x-x^2+x^5-x^6); print(f.coefficients()) # Bruno Berselli, May 12 2015

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).