%I #28 Dec 04 2023 06:44:15
%S 0,1,21,156,700,2325,6321,14896,31536,61425,111925,193116,318396,
%T 505141,775425,1156800,1683136,2395521,3343221,4584700,6188700,
%U 8235381,10817521,14041776,18030000,22920625,28870101,36054396,44670556,54938325,67101825,81431296,98224896
%N a(n) = n^2 * (n+1)^2 * (n^2+n+1) / 12.
%C Let b(n,k) = Sum_{j=0..n} (-1)^(n-j)* j^k * binomial(n,j) * binomial(n+j,j).
%C b(n,0) = 1.
%C b(n,1) = 1/1! * n * (n+1).
%C b(n,2) = 1/2! * n^2 * (n+1)^2.
%C b(n,3) = 1/3! * n^2 * (n+1)^2 * (n^2+n+1) (= 2*a(n)).
%C b(n,4) = 1/4! * n^3 * (n+1)^3 * (n^2+n+4).
%C b(n,5) = 1/5! * n^2 * (n+1)^2 * (n^6+3*n^5+13*n^4+21*n^3+18*n^2+8*n-4).
%C b(n,6) = 1/6! * n^3 * (n+1)^3 * (n^2+n+4) * (n^4+2*n^3+17*n^2+16*n-6).
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F G.f.: x * (x^4+14*x^3+30*x^2+14*x+1)/(1-x)^7.
%t a[n_] := (n*(n+1))^2 * (n^2+n+1) / 12; Array[a, 33, 0] (* _Amiram Eldar_, May 05 2021 *)
%o (PARI) {a(n) = n^2*(n+1)^2*(n^2+n+1)/12}
%o (Magma) [n^2*(n+1)^2*(n^2+n+1)/12:n in [0..32]]; // _Marius A. Burtea_, Jan 19 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 33); [0] cat (Coefficients(R!( x*(x^4+14*x^3+30*x^2+14*x+1)/(1-x)^7))); // _Marius A. Burtea_, Jan 19 2020
%Y Cf. A002378 (b(n,1)), A163102 (b(n,2)), A168178 (first differences).
%K nonn,easy
%O 0,3
%A _Seiichi Manyama_, Jan 19 2020