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%I #35 Sep 08 2022 08:45:11
%S 0,0,5,27,84,200,405,735,1232,1944,2925,4235,5940,8112,10829,14175,
%T 18240,23120,28917,35739,43700,52920,63525,75647,89424,105000,122525,
%U 142155,164052,188384,215325,245055,277760,313632,352869,395675,442260
%N a(n) = T(n)^2 - n^2, where T(n) is a triangular number.
%C a(n) is the dimension of the second Cartan power of sl(n, C), which is the irreducible representation of sl(n, C) the highest weight of which is twice that of the adjoint representation. - _Daniel J. F. Fox_, Jan 01 2006
%C Also the Harary index of the n X n rook graph. - _Eric W. Weisstein_, Jun 20 2017
%C a(n) is the dimension of the space of curvature tensors of Kähler type with vanishing Ricci trace on a Hermitian vector space of real dimension 2n. - _Daniel J. F. Fox_, Nov 21 2018
%H Vincenzo Librandi, <a href="/A085740/b085740.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Sitaramayya, <a href="https://www.ams.org/journals/tran/1973-183-00/S0002-9947-1973-0322722-1/S0002-9947-1973-0322722-1.pdf">Curvature tensors in Kaehler manifolds</a>, Transactions of the AMS, 183 (September 1973), 341-353.
%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/RookGraph.html">Rook 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^2*(n^2+2*n-3)/4.
%F a(n) = A000096(n)*A000217(n-1).
%F a(n) = 4*A173963(n+1). - _Reinhard Zumkeller_, Mar 03 2010
%F G.f.: x^2*(5+2*x-x^2)/(1-x)^5. - _Colin Barker_, Mar 17 2012
%e a(3) = T(3)^2 - 3^2 = 6^2 - 9 = 36-9 = 27.
%t Table[(n - 1) n^2 (n + 3)/4, {n, 20}] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t Table[PolygonalNumber[n]^2 - n^2, {n, 20}] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 27, 84, 200}, 20] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t CoefficientList[Series[(x (-5 - 2 x + x^2))/(-1 + x)^5, {x, 0, 20}],
%t x] (* _Eric W. Weisstein_, Jun 20 2017 *)
%o (PARI) for(n=0,50,print1(n^2*(n^2-9)/4","))
%o (Magma)[n^2*(n^2+2*n-3)/4: n in [0..40]]; // _Vincenzo Librandi_, Sep 09 2011
%Y Cf. A000096, A000217, A173963.
%K nonn,easy
%O 0,3
%A _Jon Perry_, Jul 21 2003
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