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%I #42 Mar 03 2024 17:48:22
%S 1,15,139,862,4079,15791,52450,154279,411180,1009741,2314278,5000125,
%T 10264997,20152950,38037517,69323949,122448455,210271756,351989816,
%U 575711716,921889652,1447822620,2233501928,3389114724,5064582169,7461570579,10848490675,15579077786,22115241763,31054971635,43166197978,59427633555,81077755892,109673237289,147158299390,195946638641
%N Number of ways to put n copies of 1,2,3,4 into sets.
%C Related to generalized Bell Numbers.
%C The n copies of each digit must be in different sets, and the sets must be nonempty.
%C 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.
%H Alois P. Heinz, <a href="/A175707/b175707.txt">Table of n, a(n) for n = 0..1000</a>
%H Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/babushkas.html">In How Many Ways Can You Reassemble Several Russian Dolls? </a> (2009).
%H Doron Zeilberger, <a href="http://arxiv.org/abs/0909.3453">In How many ways can you reassemble several russian dolls?</a>, arXiv:0909.3453 [math.CO], 2009.
%H Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/tokhniot/BABUSHKAS">BABUSHKAS</a>; <a href="/A165434/a165434.txt">Local copy</a>
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1).
%F 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)).
%F 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.
%F 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).
%e For n=1, the solution is the fourth term of Bell numbers A000110.
%e 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}.
%p 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));
%p seq(a(n), n=0..40);
%p seq(SeqBrnDJ(n,4)[5], n=1..6); # using the Maple package BABUSHKAS (see links)
%t 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 *)
%Y Cf. A000110, A011863, A020554, A165434.
%K nonn
%O 0,2
%A _Thotsaporn Thanatipanonda_, Dec 04 2010
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