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A092364
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a(n) = n^2*binomial(n,2).
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10
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0, 4, 27, 96, 250, 540, 1029, 1792, 2916, 4500, 6655, 9504, 13182, 17836, 23625, 30720, 39304, 49572, 61731, 76000, 92610, 111804, 133837, 158976, 187500, 219700, 255879, 296352, 341446, 391500, 446865, 507904, 574992, 648516, 728875, 816480
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OFFSET
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1,2
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COMMENTS
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Coefficient of x^2 in expansion of (1+n*x)^n.
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
Also the Harary index of the n X n rook complement graph for n != 2. - Eric W. Weisstein, Sep 14 2017
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LINKS
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Table of n, a(n) for n=1..36.
Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, Rook Complement Graph
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
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FORMULA
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a(n) = n^3*(n-1)/2. Equals A085540(n-1)/2. - Zerinvary Lajos, May 09 2007, corrected Mar 10 2011
G.f.: -x^2*(4+7*x+x^2) / (x-1)^5. - R. J. Mathar, Mar 10 2011
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
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=2} 1/a(n) = 6 - Pi^2/3 - 2*zeta(3).
Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 + 4*log(2) + 3*zeta(3)/2 - 6. (End)
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MAPLE
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A092364 := proc(n) n^3*(n-1)/2 ; end proc: # R. J. Mathar, Mar 10 2011
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MATHEMATICA
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f[n_]:=(n^4-n^3)/2; lst={}; Do[AppendTo[lst, f[n]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 04 2009 *)
Table[n^2 Binomial[n, 2], {n, 20}] (* Eric W. Weisstein, Sep 14 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 4, 27, 96, 250}, 20] (* Eric W. Weisstein, Sep 14 2017 *)
CoefficientList[Series[-((x (4 + 7 x + x^2))/(-1 + x)^5), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 14 2017 *)
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PROG
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(PARI) z(n)=n^2*binomial(n, 2); for(i=1, 40, print1(", "z(i)))
(Magma) [n^3*(n-1)/2: n in [1..50]]; // Wesley Ivan Hurt, Jun 04 2014
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CROSSREFS
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Cf. A085540.
Sequence in context: A220019 A225902 A071174 * A296316 A071175 A352331
Adjacent sequences: A092361 A092362 A092363 * A092365 A092366 A092367
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KEYWORD
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nonn,easy,changed
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AUTHOR
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Jon Perry, Mar 19 2004
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STATUS
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approved
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