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A052749
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a(n) = 2*n * Stirling2(n-1,2).
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5
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0, 0, 0, 6, 24, 70, 180, 434, 1008, 2286, 5100, 11242, 24552, 53222, 114660, 245730, 524256, 1114078, 2359260, 4980698, 10485720, 22020054, 46137300, 96468946, 201326544, 419430350, 872415180, 1811939274, 3758096328, 7784628166, 16106127300, 33285996482
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OFFSET
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0,4
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COMMENTS
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a(n) is the number of ordered set partitions of an n-set into 3 nonempty sets such that the first set contains exactly one element. a(5) = 70 since the ordered set partitions are the following: 20 of type {1},{2,3,4},{5}; 30 of type {1},{2,3},{4,5}; 20 of type {1},{2},{3,4,5}. - Enrique Navarrete, Jun 11 2023
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LINKS
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FORMULA
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E.g.f.: x*exp(x)^2 - 2*x*exp(x) + x.
Recurrence: {a(1)=0, a(2)=0, a(3)=6, (2*n^2+6*n+4)*a(n)+(-6*n-3*n^2)*a(n+1)+(n^2+n)*a(n+2)}.
O.g.f.: 2*x^3*(3-6*x+2*x^2)/((-1+x)^2*(-1+2*x)^2). - R. J. Mathar, Dec 05 2007
a(n) = Sum_{j=1..n} ( Sum_{i=2..n-1} (j+1)*2^(j-i-1) ). - Wesley Ivan Hurt, Nov 17 2014
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MAPLE
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spec := [S, {B=Set(Z, 1 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
g := taylor(exp(x)^2*x-2*x*exp(x)+x, x=0, 121): q := seq(coeff(g, x, i)*i!, i=0..120);
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MATHEMATICA
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LinearRecurrence[{6, -13, 12, -4}, {0, 0, 0, 6, 24, 70}, 40] (* Harvey P. Dale, Aug 30 2017 *)
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PROG
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(Magma) [n le 2 select 0 else n*(2^(n-1)-2): n in [0..40]]; // Vincenzo Librandi, Nov 18 2014
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001
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STATUS
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approved
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