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 A257863 Expansion of 1/(1 - x - x^2 + x^5 - x^6). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 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. = 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

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Last modified June 22 17:18 EDT 2021. Contains 345388 sequences. (Running on oeis4.)