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%I M4160 #72 Oct 11 2023 11:26:45
%S 0,1,6,24,70,165,336,616,1044,1665,2530,3696,5226,7189,9660,12720,
%T 16456,20961,26334,32680,40110,48741,58696,70104,83100,97825,114426,
%U 133056,153874,177045,202740,231136,262416,296769,334390,375480,420246,468901,521664,578760
%N a(n) = (n^4 + n^2 + 2*n)/4.
%C Number of ways to color vertices of a square using <= n colors, allowing only rotations.
%C Also product of first and last terms in n-th row of a triangle of form: row(1)= 1; row(2)= 2,3; row(3) = 4, 5, 6, ... . - _Dave Durgin_, Aug 17 2012
%D Nick Baxter, The Burnside di-lemma: combinatorics and puzzle symmetry, in Tribute to a Mathemagician, Peters, 2005, pp. 199-210.
%D M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006528/b006528.txt">Table of n, a(n) for n = 0..1000</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%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) = n*(n+1)*(n^2-n+2)/4.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Vincenzo Librandi_, Apr 30 2012
%F From _Stefano Spezia_, Oct 11 2023: (Start)
%F O.g.f.: x*(1 + x + 4*x^2)/(1 - x)^5.
%F E.g.f.: exp(x)*x*(4 + 8*x + 6*x^2 + x^3)/4. (End)
%p A006528:=-z*(1+z+4*z**2)/(z-1)**5; # _Simon Plouffe_ in his 1992 dissertation
%p a:=n->add(n+add(binomial(n,2), j=1..n),j=0..n):seq(a(n)/2, n=0..35); # _Zerinvary Lajos_, Aug 26 2008
%t Table[CycleIndex[CyclicGroup[4],t]/.Table[t[i]->n,{i,1,4}],{n,0,20}] (* _Geoffrey Critzer_, Mar 13 2011*)
%t Table[(n^4+n^2+2*n)/4,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,6,24,70},40] (* _Harvey P. Dale_, Jan 13 2019 *)
%o (Magma) I:=[0, 1, 6, 24, 70]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Apr 30 2012
%o (PARI) a(n) = n*(n+1)*(n^2-n+2)/4; /* _Joerg Arndt_, Apr 30 2012 */
%Y Row n=2 of A343095.
%Y Cf. A002817 (square colorings).
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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