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A175707
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Number of ways to put n copies of 1,2,3,4 into sets.
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2
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1, 15, 139, 862, 4079, 15791, 52450, 154279, 411180, 1009741, 2314278, 5000125, 10264997, 20152950, 38037517, 69323949, 122448455, 210271756, 351989816, 575711716, 921889652, 1447822620, 2233501928, 3389114724, 5064582169, 7461570579, 10848490675, 15579077786, 22115241763, 31054971635, 43166197978, 59427633555, 81077755892, 109673237289, 147158299390, 195946638641
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OFFSET
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0,2
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COMMENTS
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Related to generalized Bell Numbers.
The n copies of each digit must be in different sets, and the sets must be nonempty.
Other definition: Number of ways to distribute n copies of 1,2,3,4 into an arbitrary number of (nonempty) sets. Due to the nature of sets, the same digit may not be several times in the same set.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1).
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FORMULA
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a(n) = (5382*n^11 +236808*n^10 +4643760*n^9 +53507520*n^8 +402098796*n^7 +2067612624*n^6 +7421736960*n^5 +18616942080*n^4 +32101468047*n^3 +36555545268*n^2 +25131098880*n +8024016000 +7016625*(-1)^n*n^3 +84199500*(-1)^n*n^2 +359251200*(-1)^n*n +538876800*(-1)^n) / (2^11*3^7*5^2*7*11) +5/3^6*(-1)^n * (sin(n*Pi/3)/sqrt(3) +cos(n*Pi/3)).
Recurrence: a(n) -7*a(n-1) +17*a(n-2) -8*a(n-3) -36*a(n-4) +60*a(n-5) -4*a(n-6) -56*a(n-7) +22*a(n-8) +22*a(n-9) +22*a(n-10) -56*a(n-11) -4*a(n-12) +60*a(n-13) -36*a(n-14) -8*a(n-15) +17*a(n-16) -7*a(n-17) +a(n-18) = 0.
G.f.: (x^10 +8*x^9 +51*x^8 +136*x^7 +252*x^6 +300*x^5 +252*x^4 +136*x^3 +51*x^2 +8*x+1) / ((x^2+x+1)*(x+1)^4*(x-1)^12).
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EXAMPLE
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For n=1, the solution is the fourth term of Bell numbers A000110.
For n=2, one way to partition 2 copies of 1, 2 copies of 2, 2 copies of 3 and 2 copies of 4 is {1}{2}{34}{12}{34}. On the other hand {112}{34}{23}{4} is not allowed since the same numbers are in the same set {112}.
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MAPLE
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a:= n-> (5382*n^11 +236808*n^10 +4643760*n^9 +53507520*n^8 +402098796*n^7 +2067612624*n^6 +7421736960*n^5 +18616942080*n^4 +32101468047*n^3 +36555545268*n^2 +25131098880*n +8024016000 +7016625*(-1)^n*n^3 +84199500*(-1)^n*n^2 +359251200*(-1)^n*n +538876800*(-1)^n) /(2^11*3^7*5^2*7*11) +5/3^6*(-1)^n * (sin(n*Pi/3)/sqrt(3)+ cos(n*Pi/3));
seq(a(n), n=0..40);
seq(SeqBrnDJ(n, 4)[5], n=1..6); # using the Maple package BABUSHKAS (see links)
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MATHEMATICA
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LinearRecurrence[{7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1}, {1, 15, 139, 862, 4079, 15791, 52450, 154279, 411180, 1009741, 2314278, 5000125, 10264997, 20152950, 38037517, 69323949, 122448455, 210271756}, 36] (* Jean-François Alcover, Nov 13 2018 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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