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A092170 Sum of squares of alternating factorials : n!^2 - (n-1)!^2 + (n-2)!^2 - ... 1!^2. 1

%I #9 Aug 21 2016 12:57:58

%S 1,3,33,543,13857,504543,24897057,1600805343,130081089057,

%T 13038108350943,1580312813889057,227862219988670943,

%U 38547925823643969057,7561506530728353470943,1702450746193471070529057

%N Sum of squares of alternating factorials : n!^2 - (n-1)!^2 + (n-2)!^2 - ... 1!^2.

%C The height of a regular simplex (hypertetrahedron) of dimension n and with unit length edges will be h(n)=sqrt(a(n))/n!. The contents (hypervolume) will then be V(n)=V(n-1)*h(n)/n where V(1)=1.

%H Harvey P. Dale, <a href="/A092170/b092170.txt">Table of n, a(n) for n = 1..253</a>

%F a(n) = n!^2 - a(n-1), a(1)=1. - _Charles R Greathouse IV_, Oct 13 2004

%e a(3)=3!^2-a(2)=36-a(2);

%e a(2)=2!^2-a(1)=4-a(1)=3-1=3 ->

%e a(3)=36-3=33.

%t a[n_] := Sum[(-1)^j*((n - j)!)^2, {j, 0, n - 1}]

%t Module[{nn=20,fctrls}, fctrls=(Range[nn]!)^2;Table[Total[Times@@@ Partition[ Riffle[Reverse[Take[fctrls,n]],{1,-1},{2,-1,2}],2]], {n, nn}]] (* _Harvey P. Dale_, Aug 21 2016 *)

%Y Cf. A005165, A055546.

%K easy,nonn

%O 1,2

%A Christer Mauritz Blomqvist (MauritzTortoise(AT)hotmail.com), Apr 01 2004

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Last modified June 21 11:14 EDT 2024. Contains 373544 sequences. (Running on oeis4.)