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A051743
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a(n)=(1/24)*n*(n+5)*(n^2+n+6).
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3
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2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, 1377, 1833, 2394, 3075, 3892, 4862, 6003, 7334, 8875, 10647, 12672, 14973, 17574, 20500, 23777, 27432, 31493, 35989, 40950, 46407, 52392, 58938, 66079, 73850, 82287, 91427, 101308, 111969, 123450
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| This is exactly the number of directed column-convex polyominoes
Let A be the Hessenberg n by n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-3)=coeff(charpoly(A,x),x^(n-4)). [From Milan R. Janjic (agnus(AT)blic.net), Jan 24 2010]
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LINKS
| Harvey P. Dale, Table of n, a(n) for n = 1..1000
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FORMULA
| a(n)=binomial(n+3, n-1) + binomial(n, n-1) = binomial(n+3, 4) + binomial(n, 1), n > 0.
a(1)=2, a(2)=7, a(3)=18, a(4)=39, a(5)=75, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a (n-5) [From Harvey P. Dale, Nov 29 2011]
G.f.: (x^3-3*x^2+3*x-2)/(x-1)^5 [From Harvey P. Dale, Nov 29 2011]
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MATHEMATICA
| Table[(n(n+5)(n^2+n+6))/24, {n, 50}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {2, 7, 18, 39, 75}, 50]
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CROSSREFS
| Cf. A005435, A006027, A105450.
Sequence in context: A077131 A176854 A086741 * A054111 A192955 A055503
Adjacent sequences: A051740 A051741 A051742 * A051744 A051745 A051746
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KEYWORD
| easy,nonn
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AUTHOR
| Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999
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