cp's OEIS Frontend

A062027 a(1) = a(2) = a(3) = 1 and a(n) = 24*binomial(n+1, 5) + n*(n^2 - n + 6) for n > 3.

%I A062027 #24 Dec 07 2024 13:39:49
%S A062027 1,1,1,96,274,720,1680,3520,6750,12048,20284,32544,50154,74704,108072,
%T A062027 152448,210358,284688,378708,496096,640962,817872,1031872,1288512,
%U A062027 1593870,1954576,2377836,2871456,3443866,4104144,4862040,5728000
%N A062027 a(1) = a(2) = a(3) = 1 and a(n) = 24*binomial(n+1, 5) + n*(n^2 - n + 6) for n > 3.
%C A062027 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 A062027 Harry J. Smith, <a href="/A062027/b062027.txt">Table of n, a(n) for n=1..1000</a>
%H A062027 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A062027 a(n) = (n+1)*(n)*(n-1)*(n-2)*(n-3)/5 + n*(n^2 - n + 6), for n>3.
%F A062027 From _G. C. Greubel_, May 05 2022: (Start)
%F A062027 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 A062027 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 A062027 a(5) = 1*2*3*4 + 2*3*4*5 + 3*4*5*1 + 4*5*1*2 + 5*1*2*3 = 274.
%t A062027 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 A062027 (PARI) a(n) = { if (n<=3, 1, (n+1)*n*(n-1)*(n-2)*(n-3)/5 +n*(n^2-n+6)) } \\ _Harry J. Smith_, Jul 30 2009
%o A062027 (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 A062027 nonn,easy
%O A062027 1,4
%A A062027 _Amarnath Murthy_, Jun 02 2001
%E A062027 More terms from _Jason Earls_, Jun 07 2001
%E A062027 Term a(4) corrected by _Harry J. Smith_, Jul 30 2009
%E A062027 Name changed by _G. C. Greubel_, May 05 2022