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A091344
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a(n) = 2*3^n - 3*2^n + 1.
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10
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0, 1, 7, 31, 115, 391, 1267, 3991, 12355, 37831, 115027, 348151, 1050595, 3164071, 9516787, 28599511, 85896835, 257887111, 774054547, 2322950071, 6970423075, 20914414951, 62749536307, 188261191831, 564808741315, 1694476555591
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OFFSET
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0,3
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COMMENTS
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Starting with offset 1 = binomial transform of A068293: (1, 6, 18, 42, 90, ...) and double binomial transform of (1, 5, 7, 5, 7, 5, ...). - Gary W. Adamson, Jan 13 2009
Number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a subset of the other. - For the case that one of these subsets is a proper subset of the other see a(n+1) in A260217. - If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). - Manfred Boergens, Aug 02 2023
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n} i!*i^2*Stirling2(n,i)*(-1)^(n-i).
From Christian Ballot via R. K. Guy, Jan 13 2009: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3);
G.f.: x*(1+x)/((1-x)*(2-x)*(3-x)). (End)
E.g.f.: exp(x)*(1 - 3*exp(x) + 2*exp(2*x)). - Stefano Spezia, May 18 2024
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MAPLE
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a:=n->sum((3^(n-j-1)-2^(n-2-j))*12, j=0..n): seq(a(n), n=-1..24); # Zerinvary Lajos, Feb 11 2007
with (combinat):a:=n->stirling2(n, 3)+stirling2(n+1, 3): seq(a(n), n=1..26); # Zerinvary Lajos, Oct 07 2007
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MATHEMATICA
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Table[Sum[i!i^2 StirlingS2[n, i](-1)^(n - i), {i, 1, n}], {n, 0, 30}]
Table[2*3^n-3*2^n+1, {n, 0, 30}] (* or *) LinearRecurrence[{6, -11, 6}, {0, 1, 7}, 30] (* Harvey P. Dale, Dec 31 2013 *)
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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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Mario Catalani (mario.catalani(AT)unito.it), Jan 01 2004
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EXTENSIONS
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STATUS
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approved
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