A092364 a(n) = n^2*binomial(n,2).

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

Offset: 1

Author

Jon Perry, Mar 19 2004

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A085540.

Programs

Magma

[n^3*(n-1)/2: n in [1..50]]; // Wesley Ivan Hurt, Jun 04 2014

Maple

A092364 := proc(n) n^3*(n-1)/2 ; end proc: # R. J. Mathar, Mar 10 2011

Mathematica

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 *)

PARI

z(n)=n^2*binomial(n,2); for(i=1,40,print1(","z(i)))

Formula

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)

E.g.f.: exp(x)*x^2*(4 + 5*x + x^2)/2. - Stefano Spezia, Jun 10 2023