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A019582 n*(n-1)^3/2. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

a(n+2) = denominator(2/((n+2)*(n+1)^3)). - Stephen Crowley, Jun 28 2009

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..680

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( 1/A019582(j), j=2..infinity ) = 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

EXAMPLE

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

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 *)

PROG

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

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

CROSSREFS

Cf. A000217, A000290, A092364.

A row or column of A132191.

Sequence in context: A060785 A059986 A088941 * A025204 A005549 A124858

Adjacent sequences:  A019579 A019580 A019581 * A019583 A019584 A019585

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 11 1996

STATUS

approved

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Last modified July 23 22:49 EDT 2019. Contains 325278 sequences. (Running on oeis4.)