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A132117
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Binomial transform of [1, 7, 17, 17, 6, 0, 0, 0,...].
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4
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1, 8, 32, 90, 205, 406, 728, 1212, 1905, 2860, 4136, 5798, 7917, 10570, 13840, 17816, 22593, 28272, 34960, 42770, 51821, 62238, 74152, 87700, 103025, 120276, 139608, 161182, 185165, 211730, 241056, 273328, 308737, 347480, 389760, 435786, 485773, 539942, 598520
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Equals row sums of triangle A178067 [From Gary W. Adamson, May 18 2010]
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
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FORMULA
| Let M = the infinite lower triangular matrix of the natural numbers: [1; 2,3; 4,5,6;...]; and V = [1, 2, 3,...]. Then M*V = A132117.
O.g.f.: -x(1+x)(2x+1)/(-1+x)^5. - R. J. Mathar, Apr 02 2008
a(n) = (4*n+6*n^2+8*n^3+6*n^4)/24 [From Alois P. Heinz, Aug 07 2008]
a(n) = A000217(n)^2 - [A000217(n-1)+ ... +6+3+1] = A000217(n)^2 - A000292(n-1) = n*(n+1)*(3*n^2+n+2)/12 [From Bruno Berselli, May 01 2010]
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EXAMPLE
| a(3) = 32 = (1, 2, 1) dot (1, 7, 17) = (1 + 14 + 17).
a(5) = 15^2 - (10+6+3+1) = 225 - 20 = 5*(5+1)*(3*5^2+5+2)/12 = 205 [From Bruno Berselli, May 01 2010]
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MAPLE
| a:= n-> (Matrix([[0, 0, 2, 13, 46]]). Matrix(5, (i, j)-> if (i=j-1) then 1 elif j=1 then [5, -10, 10, -5, 1][i] else 0 fi)^n)[1, 1]: seq (a(n), n=1..29); # Alois P. Heinz, Aug 07 2008
a:= n-> (4+(6+(8+6*n)*n)*n)*n/24: seq (a(n), n=1..40); # Alois P. Heinz, Aug 07 2008
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PROG
| (PARI) a(n) = (4*n+6*n^2+8*n^3+6*n^4)/24 \\ Charles R Greathouse IV, Sep 03 2011
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CROSSREFS
| Cf. A178067 [From Gary W. Adamson, May 18 2010]
Sequence in context: A008412 A014819 A033155 * A159941 A053348 A019256
Adjacent sequences: A132114 A132115 A132116 * A132118 A132119 A132120
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KEYWORD
| nonn,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2007
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 02 2008
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