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%I #25 Sep 08 2022 08:46:12
%S 1,1,2,3,5,7,12,18,29,45,72,112,178,279,441,693,1094,1721,2714,4273,
%T 6735,10607,16715,26329,41485,65352,102965,162209,255560,402613,
%U 634306,999306,1574368,2480323,3907638,6156268,9698906,15280112,24073063,37925860,59750293
%N Expansion of 1/(1 - x - x^2 + x^5 - x^6).
%C 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.)
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,1).
%F G.f.: 1/(1-x-x^2+x^5-x^6).
%F a(n) = a(n-1) + a(n-2) - a(n-5) + a(n-6).
%e 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)
%t 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 *)
%t 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 *)
%o (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
%o (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
%K nonn,easy
%O 0,3
%A _David Neil McGrath_, May 11 2015
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