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%I #57 Feb 13 2024 20:04:49
%S 1,11,46,130,295,581,1036,1716,2685,4015,5786,8086,11011,14665,19160,
%T 24616,31161,38931,48070,58730,71071,85261,101476,119900,140725,
%U 164151,190386,219646,252155,288145,327856,371536,419441,471835,528990,591186
%N a(n) = n*(n + 1)*(2*n^2 + 2*n - 1)/6.
%C 4-dimensional analog of centered polygonal numbers.
%C Partial sums of A000447. - _Zak Seidov_, May 19 2006
%C From _Johannes W. Meijer_, Jun 27 2009: (Start)
%C Equals the absolute values of the coefficients that precede the a(n-1) factors of the recurrence relations RR(n) of A162011.
%C This sequence enabled the analysis of A162012 and A162013. (End)
%C Equals the number of integer quadruples (x,y,z,w) such that min(x,y) < min(z,w), max(x,y) < max(z,w), and 0 <= x,y,z,w <= n. - _Andrew Woods_, Apr 21 2014
%C For n>3 a(n)=twice the area of an irregular quadrilateral with vertices at the points (C(n,4),C(n+1,4)), (C(n+1,4),C(n+2,4)), (C(n+2,4),C(n+3,4)), and (C(n+3,4),C(n+4,4)). - _J. M. Bergot_, Jun 14 2014
%H Delbert L. Johnson, <a href="/A006324/b006324.txt">Table of n, a(n) for n = 1..20000</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(n) = 8*C(n + 2, 4) + C(n + 1, 2).
%F a(n) = (Sum_{k=1..n} k^5) / (Sum_{k=1..n} k) = A000539(n) / A000217(n). - _Alexander Adamchuk_, Apr 12 2006
%F From _Johannes W. Meijer_, Jun 27 2009: (Start)
%F Recurrence relation 0 = Sum_{k=0..5} (-1)^k*binomial(5,k)*a(n-k).
%F G.f.: (1+6*z+z^2)/(1-z)^5. (End)
%F a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - _Wesley Ivan Hurt_, May 02 2021
%F Sum_{n>=1} 1/a(n) = 6 + 2*sqrt(3)*Pi*tan(sqrt(3)*Pi/2). - _Amiram Eldar_, Aug 23 2022
%F a(n) = A053134(n-1) - 4*A002415(n). - _Yasser Arath Chavez Reyes_, Feb 12 2024
%p A006324:=n->n*(n + 1)*(2*n^2 + 2*n - 1)/6; seq(A006324(n), n=1..30); # _Wesley Ivan Hurt_, Jun 14 2014
%t Table[Sum[k^5,{k,n}]/Sum[k,{k,n}], {n,40}] (* _Alexander Adamchuk_, Apr 12 2006 *)
%o (Magma) [ n*(n + 1)*(2*n^2 + 2*n - 1)/6 : n in [1..30] ]; // _Wesley Ivan Hurt_, Jun 14 2014
%Y Cf. A000447, A000539, A000217, A002415, A053134.
%Y Cf. A162011, A162012, a(n-2), and A162013, a(n-3). - _Johannes W. Meijer_, Jun 27 2009
%K nonn,easy
%O 1,2
%A Albert Rich (Albert_Rich(AT)msn.com), Jun 14 1998
%E Simpler definition from _Alexander Adamchuk_, Apr 12 2006
%E More terms from _Zak Seidov_
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