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A257863
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Expansion of 1/(1 - x - x^2 + x^5 - x^6).
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1
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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
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OFFSET
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0,3
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COMMENTS
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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.)
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LINKS
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FORMULA
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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).
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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