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

%I A257863 #25 Sep 08 2022 08:46:12
%S A257863 1,1,2,3,5,7,12,18,29,45,72,112,178,279,441,693,1094,1721,2714,4273,
%T A257863 6735,10607,16715,26329,41485,65352,102965,162209,255560,402613,
%U A257863 634306,999306,1574368,2480323,3907638,6156268,9698906,15280112,24073063,37925860,59750293
%N A257863 Expansion of 1/(1 - x - x^2 + x^5 - x^6).
%C A257863 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 A257863 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,1).
%F A257863 G.f.: 1/(1-x-x^2+x^5-x^6).
%F A257863 a(n) = a(n-1) + a(n-2) - a(n-5) + a(n-6).
%e A257863 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 A257863 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 A257863 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 A257863 (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 A257863 (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 A257863 nonn,easy
%O A257863 0,3
%A A257863 _David Neil McGrath_, May 11 2015