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A180118
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a(n) = Sum_{k=1..n} (k+2)!/k! = Sum_{k=1..n} (k+2)*(k+1).
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1
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0, 6, 18, 38, 68, 110, 166, 238, 328, 438, 570, 726, 908, 1118, 1358, 1630, 1936, 2278, 2658, 3078, 3540, 4046, 4598, 5198, 5848, 6550, 7306, 8118, 8988, 9918, 10910, 11966, 13088, 14278, 15538, 16870, 18276, 19758, 21318, 22958, 24680, 26486, 28378, 30358
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OFFSET
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0,2
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COMMENTS
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In general, sequences of the form a(n) = sum((k+x+2)!/(k+x)!,k=1..n) have a closed form a(n) = n*(11+12*x+3*x^2+3*x*n+6*n+n^2)/3.
The minimal number of multiplications (using schoolbook method) needed to compute the matrix chain product of a sequence of n+1 matrices having dimensions 1 X 2, 2 X 3, ..., (n+1) X (n+2), respectively. - Alois P. Heinz, Jan 27 2017
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LINKS
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B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
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FORMULA
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a(n) = +4*a(n-1)-6*a(n-2)+4*a(n-3)-1*a(n-4) for n>=4.
a(n) = n*(n^2+6*n+11)/3.
G.f.: 2*x*(3-3*x+x^2)/(1-x)^4. [corrected by Georg Fischer, May 10 2019]
Sum(a(k), k=0..n) = 2*A005718(n) for n>0. (End)
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MATHEMATICA
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CoefficientList[Series[2*x*(3 - 3*x + x^2)/(1 - x)^4, {x, 0, 50}], x] (* Vaclav Kotesovec, May 10 2019 *)
Table[Sum[(k+1)(k+2), {k, n}], {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 6, 18, 38}, 50] (* Harvey P. Dale, Apr 21 2020 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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