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%I #14 Sep 08 2022 08:45:32
%S 0,60,396,1386,3570,7650,14490,25116,40716,62640,92400,131670,182286,
%T 246246,325710,423000,540600,681156,847476,1042530,1269450,1531530,
%U 1832226,2175156,2564100,3003000,3495960,4047246,4661286,5342670
%N Sums of the products of n consecutive triples of numbers.
%H G. C. Greubel, <a href="/A135037/b135037.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(1) = 0*1*2, a(2) = 0*1*2 + 3*4*5, ..., a(n) = 0*1*2 + 3*4*5 + 6*7*8 + ... + (2n-1)*(2n)*(2n+1).
%F a(n) = (27*n^4 - 18*n^3 - 15*n^2 + 6*n)/4.
%F From _R. J. Mathar_, Feb 14 2008: (Start)
%F O.g.f.: 6*x^2*(10+16*x+x^2)/(1-x)^5.
%F a(n) = 6*A024391(n-1). (End)
%F E.g.f.: (3/4)*x^2*(40 + 48*x + 9*x^2)*exp(x). - _G. C. Greubel_, Sep 17 2016
%e For n = 3, the sum of the first 3 triples is 0*1*2+3*4*5+6*7*8 =396, the 3rd entry in the sequence.
%t Table[(27 n^4 - 18 n^3 - 15 n^2 + 6 n)/4, {n, 1, 50}] (* or *) LinearRecurrence[{5,-10,10,-5,1}, {0, 60, 396, 1386, 3570}, 25] (* _G. C. Greubel_, Sep 17 2016 *)
%o (PARI) sumprod3(n) = { local(x,s=0); forstep(x=0,n,3, s+=x*(x+1)*(x+2); print1(s",") ) }
%o (Magma) [(27*n^4-18*n^3-15*n^2+6*n)/4: n in [1..40]]; // _Vincenzo Librandi_, Sep 18 2016
%K nonn,easy
%O 1,2
%A _Cino Hilliard_, Feb 10 2008
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