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%I #53 Sep 08 2022 08:45:54
%S 0,6,18,38,68,110,166,238,328,438,570,726,908,1118,1358,1630,1936,
%T 2278,2658,3078,3540,4046,4598,5198,5848,6550,7306,8118,8988,9918,
%U 10910,11966,13088,14278,15538,16870,18276,19758,21318,22958,24680,26486,28378,30358
%N a(n) = Sum_{k=1..n} (k+2)!/k! = Sum_{k=1..n} (k+2)*(k+1).
%C 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.
%C This sequence is related to A033487 by A033487(n) = n*a(n)-sum(a(i), i=0..n-1). - _Bruno Berselli_, Jan 24 2011
%C 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
%H Vincenzo Librandi, <a href="/A180118/b180118.txt">Table of n, a(n) for n = 0..1000</a>
%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Matrix_chain_multiplication">Matrix chain multiplication</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Matrix_multiplication_algorithm#Iterative_algorithm">Schoolbook matrix multiplication</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = +4*a(n-1)-6*a(n-2)+4*a(n-3)-1*a(n-4) for n>=4.
%F a(n) = n*(n^2+6*n+11)/3.
%F From _Bruno Berselli_, Jan 24 2011: (Start)
%F G.f.: 2*x*(3-3*x+x^2)/(1-x)^4. [corrected by _Georg Fischer_, May 10 2019]
%F Sum(a(k), k=0..n) = 2*A005718(n) for n>0. (End)
%t f[n_]:=n*(n^2 + 6 n + 11)/3; f[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 10 2011*)
%t CoefficientList[Series[2*x*(3 - 3*x + x^2)/(1 - x)^4, {x, 0, 50}], x] (* _Vaclav Kotesovec_, May 10 2019 *)
%t 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 *)
%o (Magma) [n*(n^2+6*n+11)/3: n in [0..45]]; // _Vincenzo Librandi_, Jun 15 2011
%Y Cf. A005718, A033487, A050534.
%K nonn,easy
%O 0,2
%A _Gary Detlefs_, Aug 10 2010
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