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A114725
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The first entry of the vector v[n]=Mv[n-1], where M is the 6 X 6 matrix [[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 5, 10, 10, 5, 1]] and v[0] is the column vector [0, 1, 1, 2, 3, 5].
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1
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0, 1, 1, 2, 3, 5, 55, 136, 502, 1799, 6247, 21902, 76882, 269498, 944895, 3313259, 11617291, 40733828, 142826195, 500794808, 1755948163, 6156922147, 21588159423, 75695064671, 265411365121, 930618039799
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OFFSET
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0,4
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COMMENTS
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Characteristic polynomial of the matrix M is x^6-(x+1)^5.
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LINKS
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FORMULA
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Recurrence relation: a(n)=a(n-1)+5a(n-2)+10a(n-3)+10a(n-4)+5a(n-5)+a(n-6) for n>=6; a(0)=0,a(1)=a(2)=1,a(3)=2,a(4)=3,a(5)=5.
O.g.f.: x*(2*x+1)*(14*x^3+2*x-1)/(-1+x+5*x^2+10*x^3+10*x^4+5*x^5+x^6) . - R. J. Mathar, Dec 05 2007
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MAPLE
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a[0]:=0:a[1]:=1:a[2]:=1:a[3]:=2:a[4]:=3:a[5]:=5: for n from 6 to 25 do a[n]:=a[n-1]+5*a[n-2]+10*a[n-3]+10*a[n-4]+5*a[n-5]+a[n-6] od: seq(a[n], n=0..25);
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MATHEMATICA
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M = {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 4, 6, 4, 1}}; w[0] = {0, 1, 1, 2, 3}; w[n_] := w[n] = M.w[n - 1] a = Flatten[Table[w[n][[1]], {n, 0, 25}]]
LinearRecurrence[{1, 5, 10, 10, 5, 1}, {0, 1, 1, 2, 3, 5}, 30] (* Harvey P. Dale, Jun 15 2014 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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