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%I #20 May 24 2022 02:32:25
%S 1,1,1,96,274,720,1680,3520,6750,12048,20284,32544,50154,74704,108072,
%T 152448,210358,284688,378708,496096,640962,817872,1031872,1288512,
%U 1593870,1954576,2377836,2871456,3443866,4104144,4862040,5728000
%N a(1) = a(2) = a(3) = 1 and a(n) = 24*binomial(n+1, 5) + n*(n^2 - n + 6) for n > 3.
%C Previous name: a(1) = a(2) = a(3) = 1; and for n>3 a(n) = 1*2*3*4 + 2*3*4*5 + 3*4*5*6 + ... + (n-1)*n*1*2 + n*1*2*3, the sum of the cyclic product of terms taken four at a time, final term being n*1*2*3 = 6n.
%H Harry J. Smith, <a href="/A062027/b062027.txt">Table of n, a(n) for n=1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = (n+1)*(n)*(n-1)*(n-2)*(n-3)/5 + n*(n^2 - n + 6), for n>3.
%F From _G. C. Greubel_, May 05 2022: (Start)
%F G.f.: -5*x*(1 + 3*x + 7*x^2) + 2*x*(3 - 10*x + 15*x^2 + 4*x^4)/(1-x)^6.
%F E.g.f.: (1/5)*x*(30 + 10*x + 5*x^2 + 5*x^3 + x^4)*exp(x) - (5/6)*x*(6 + 9*x + 7*x^2). (End)
%e a(5) = 1*2*3*4 + 2*3*4*5 + 3*4*5*1 + 4*5*1*2 + 5*1*2*3 = 274.
%t Table[24*Binomial[n+1,5] +n*(n^2-n+6) -5*(2^n-1)*Boole[n<4], {n,40}] (* _G. C. Greubel_, May 05 2022 *)
%o (PARI) { a=1; for(n=1, 100, if (n>3, a=(n+1)*n*(n-1)*(n-2)*(n-3)/5 +n*(n^2-n+6)); write("b062027.txt", n, " ", a) ) } \\ _Harry J. Smith_, Jul 30 2009
%o (SageMath) [24*binomial(n+1,5) +n*(n^2-n+6) -5*(2^n-1)*bool(n<4) for n in (1..40)] # _G. C. Greubel_, May 05 2022
%K nonn,easy
%O 1,4
%A _Amarnath Murthy_, Jun 02 2001
%E More terms from _Jason Earls_, Jun 07 2001
%E Term a(4) corrected by _Harry J. Smith_, Jul 30 2009
%E Name changed by _G. C. Greubel_, May 05 2022
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