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a(n) = n*(n - 1)^3/2.
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%I #68 Jun 10 2023 15:55:35

%S 0,0,1,12,54,160,375,756,1372,2304,3645,5500,7986,11232,15379,20580,

%T 27000,34816,44217,55404,68590,84000,101871,122452,146004,172800,

%U 203125,237276,275562,318304,365835,418500,476656,540672,610929,687820,771750,863136

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

%C 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

%C 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

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

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A019582/b019582.txt">Table of n, a(n) for n = 0..680</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n+1) = Sum_{k=0..n} n^2(n-k) = n^3*(n+1)/2. - _Paul Barry_, Sep 02 2003

%F a(n+1) = A000290(n) * A000217(n). - _Zerinvary Lajos_, Jan 20 2007

%F 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

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

%F a(1 - n) = A092364(n). - _Michael Somos_, Jun 09 2014

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

%F E.g.f.: exp(x)*x^2*(1 + 3*x + x^2)/2. - _Stefano Spezia_, Jun 10 2023

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

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

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

%o (Magma) [n*(n-1)^3/2: n in [0..60]]; // _Vincenzo Librandi_, Apr 26 2011

%o (PARI) a(n)=n*(n-1)^3/2 \\ _Charles R Greathouse IV_, Feb 27 2017

%Y Cf. A000217, A000290, A092364.

%Y A row or column of A132191.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_, Dec 11 1996