A019582 a(n) = n*(n - 1)^3/2.

0, 0, 1, 12, 54, 160, 375, 756, 1372, 2304, 3645, 5500, 7986, 11232, 15379, 20580, 27000, 34816, 44217, 55404, 68590, 84000, 101871, 122452, 146004, 172800, 203125, 237276, 275562, 318304, 365835, 418500, 476656, 540672, 610929, 687820, 771750, 863136

Offset: 0

Author

N. J. A. Sloane, Dec 11 1996

Keywords

• nonn
• easy

Comments

a(n) = n(n-1)^3/2 is half the number of colorings of 4 points on a line with n colors. - R. H. Hardin, Feb 23 2002

n^2*n(n+1)/2: a(n+1) = product of n-th triangular number and n-th square number. E.g., a(4)=6*9=54. - Alexandre Wajnberg, Dec 18 2005

Also, the number of ways to place two dominoes horizontally in different rows on an n X n chessboard. - Ralf Stephan, Jun 09 2014

a(n) is the second Zagreb index of the complete graph K[n]. The second Zagreb index of a simple connected graph g is the sum of the degree products d(i)d(j) over all edges ij of g. - Emeric Deutsch, Nov 07 2016

a(n+1) is the number of inequivalent 2 X 2 matrices with entries in {1,2,3,...,n} when a matrix and its transpose are considered equivalent. - David Nacin, Feb 27 2017

Examples

			G.f. = x^2 + 12*x^3 + 54*x^4 + 160*x^5 + 375*x^6 + 756*x^7 + 1372*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..680
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A000290, A092364.

A row or column of A132191.

Programs

Magma

[n*(n-1)^3/2: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

Maple

f := n->n*(n-1)^3/2; seq(f(n), n=0..50);

Mathematica

f[n_]:=n*(n-1)^3/2; Table[f[n], {n,0,4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2010 *)

PARI

a(n)=n*(n-1)^3/2 \\ Charles R Greathouse IV, Feb 27 2017

Formula

a(n+1) = Sum_{k=0..n} n^2(n-k) = n^3*(n+1)/2. - Paul Barry, Sep 02 2003

a(n+1) = A000290(n) * A000217(n). - Zerinvary Lajos, Jan 20 2007

Sum_{j>=2} 1/a(j) = hypergeom([1, 1, 1, 1], [2, 2, 3], 1) = 2 - 2*zeta(2) + 2*zeta(3). - Stephen Crowley, Jun 28 2009

G.f.: -x^2*(4*x^2 + 7*x + 1)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

a(1 - n) = A092364(n). - Michael Somos, Jun 09 2014

Sum_{n>=2} (-1)^n/a(n) = 3*zeta(3)/2 - zeta(2) + 4*log(2) - 2. - Amiram Eldar, Sep 11 2022

E.g.f.: exp(x)*x^2*(1 + 3*x + x^2)/2. - Stefano Spezia, Jun 10 2023