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%I #52 May 29 2022 03:08:21
%S 0,3,24,100,300,735,1568,3024,5400,9075,14520,22308,33124,47775,67200,
%T 92480,124848,165699,216600,279300,355740,448063,558624,690000,845000,
%U 1026675,1238328,1483524,1766100,2090175
%N a(n) = n*(n+1)^2*(n+2)^2/12.
%H Vincenzo Librandi, <a href="/A004282/b004282.txt">Table of n, a(n) for n = 0..10000</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) = C(2+n, 2)*C(2+n, 3) = A000217(n+1)*A000292(n). - _Zerinvary Lajos_, Jan 10 2006
%F a(n-1) = Sum_{1 <= x_1, x_2 <= n} x_1*(det V(x_1,x_2))^2 = Sum_{1 <= i,j <= n} i*(i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - _Peter Bala_, Sep 21 2007
%F G.f.: x*(3+6*x+x^2)/(1-x)^6. - _Colin Barker_, Feb 09 2012
%F a(n) = Sum_{k=0..n} Sum_{i=0..n} (n-i+1) * C(k+1,k-1). - _Wesley Ivan Hurt_, Sep 21 2017
%F a(n) = A004302(n+1) - A000537(n+1). - _J. M. Bergot_, Mar 28 2018
%F From _Amiram Eldar_, May 29 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 30 - 3*Pi^2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 24*log(2) + 12. (End)
%p a:= n-> binomial(2+n, 2)*binomial(2+n, 3): seq(a(n), n=0..31); # _Zerinvary Lajos_, Apr 26 2007
%t Table[n*(n+1)^2*(n+2)^2/12,{n,0,40} (* _Vincenzo Librandi_, Feb 09 2012 *)
%o (Magma) [n*(n+1)^2*(n+2)^2/12: n in [0..50]]; // _Vincenzo Librandi_, Feb 09 2012
%o (PARI) a(n) = binomial(n+2,2)*binomial(n+2,3); \\ _Charles R Greathouse IV_, Feb 09 2012
%Y Cf. A006542, A107891, A114242.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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