OFFSET
1,2
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).
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jon Perry, Mar 19 2004
STATUS
approved