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 A175707 Number of ways to put n copies of 1,2,3,4 into sets. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Doron Zeilberger, In How Many Ways Can You Reassemble Several Russian Dolls? (2009). Doron Zeilberger, In How many ways can you reassemble several russian dolls?, arXiv:0909.3453 Doron Zeilberger BABUSHKAS FORMULA 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). EXAMPLE 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}. MAPLE 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) CROSSREFS Cf. A000110, A011863, A020554, A165434. Sequence in context: A126536 A030056 A225978 * A123955 A027802 A133716 Adjacent sequences:  A175704 A175705 A175706 * A175708 A175709 A175710 KEYWORD nonn AUTHOR Thotsaporn Thanatipanonda, Dec 04 2010 STATUS approved

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