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%I #46 Jun 10 2023 20:56:26
%S 0,4,27,96,250,540,1029,1792,2916,4500,6655,9504,13182,17836,23625,
%T 30720,39304,49572,61731,76000,92610,111804,133837,158976,187500,
%U 219700,255879,296352,341446,391500,446865,507904,574992,648516,728875,816480
%N a(n) = n^2*binomial(n,2).
%C Coefficient of x^2 in expansion of (1+n*x)^n.
%C For n>3, a(n) is twice the area of a triangle with vertices at points (C(n-1,3),C(n,3)), (C(n,3),C(n+1,3)), and (C(n+1,3),C(n+2,3)). - _J. M. Bergot_, Jun 05 2014
%C Also the Harary index of the n X n rook complement graph for n != 2. - _Eric W. Weisstein_, Sep 14 2017
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HararyIndex.html">Harary Index</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</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) = n^3*(n-1)/2. Equals A085540(n-1)/2. - _Zerinvary Lajos_, May 09 2007, corrected Mar 10 2011
%F G.f.: -x^2*(4+7*x+x^2) / (x-1)^5. - _R. J. Mathar_, Mar 10 2011
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Eric W. Weisstein_, Sep 14 2017
%F From _Amiram Eldar_, May 14 2022: (Start)
%F Sum_{n>=2} 1/a(n) = 6 - Pi^2/3 - 2*zeta(3).
%F Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 + 4*log(2) + 3*zeta(3)/2 - 6. (End)
%F E.g.f.: exp(x)*x^2*(4 + 5*x + x^2)/2. - _Stefano Spezia_, Jun 10 2023
%p A092364 := proc(n) n^3*(n-1)/2 ; end proc: # _R. J. Mathar_, Mar 10 2011
%t f[n_]:=(n^4-n^3)/2; lst={};Do[AppendTo[lst,f[n]],{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 04 2009 *)
%t Table[n^2 Binomial[n, 2], {n, 20}] (* _Eric W. Weisstein_, Sep 14 2017 *)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 4, 27, 96, 250}, 20] (* _Eric W. Weisstein_, Sep 14 2017 *)
%t CoefficientList[Series[-((x (4 + 7 x + x^2))/(-1 + x)^5), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 14 2017 *)
%o (PARI) z(n)=n^2*binomial(n,2); for(i=1,40,print1(","z(i)))
%o (Magma) [n^3*(n-1)/2: n in [1..50]]; // _Wesley Ivan Hurt_, Jun 04 2014
%Y Cf. A085540.
%K nonn,easy
%O 1,2
%A _Jon Perry_, Mar 19 2004
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