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%I #91 Nov 26 2023 06:55:34
%S 1,7,27,76,175,351,637,1072,1701,2575,3751,5292,7267,9751,12825,16576,
%T 21097,26487,32851,40300,48951,58927,70357,83376,98125,114751,133407,
%U 154252,177451,203175,231601,262912,297297,334951,376075,420876,469567
%N a(n) = n^2*(n^2+3)/4.
%C Previous definition was: Consider a figure like this <> (a squashed square, symmetric about both axes); each side is given 1 of n colors; a(n) = number of possibilities, allowing turning over.
%C Also number of 2 X 2 matrices with entries mod n, up to row and column permutation. Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively. - _Vladeta Jovovic_, Nov 04 2000
%C Also, if a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 08 2007
%H Harvey P. Dale, <a href="/A039623/b039623.txt">Table of n, a(n) for n = 1..1000</a>
%H Jean-Paul Delahaye, <a href="http://www.pourlascience.fr/ewb_pages/a/article-le-miraculeux-lemme-de-burnside-19753.php">Le miraculeux "lemme de Burnside"</a>, pp. 145-6 in 'Pour la Science' (French edition of 'Scientific American'), No. 350, December 2006, Paris.
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F From _Harvey P. Dale_, Oct 01 2011: (Start)
%F G.f.: (1 + 2*x + 2*x^2 + x^3)/(1 - x)^5.
%F a(1)=1, a(2)=7, a(3)=27, a(4)=76, a(5)=175; for n>5, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
%F E.g.f.: x*(4 + 10*x + 6*x^2 + x^3)*exp(x)/4. - _Ilya Gutkovskiy_, Apr 16 2016
%F a(n) = t(n-1)*t(n) + t(n-1) + t(n) where t=A000217. - _J. M. Bergot_, Apr 16 2016
%F a(n) = A000217(n)^2 - n*A000217(n-1). - _Bruno Berselli_, Feb 14 2017
%F a(n) = T(T(n-1)) + T(T(n)) where T(n) = A000217(n). - _Charlie Marion_, Feb 09 2023
%F Sum_{n>=1} 1/a(n) = 2*(1 + Pi^2 - sqrt(3)*Pi*coth(sqrt(3)*Pi))/9. - _Amiram Eldar_, Feb 13 2023
%F a(n) = binomial(n,2)*binomial(n+1,2) + n^2 = A006011(n) + A000290(n). - _Detlef Meya_, Nov 23 2023
%p A039623:=n->n^2*(n^2+3)/4: seq(A039623(n), n=1..50); # _Wesley Ivan Hurt_, Dec 26 2016
%t Table[(n^2 (n^2+3))/4,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,7,27,76,175},40] (* _Harvey P. Dale_, Oct 01 2011 *)
%o (PARI) Vec((-1-2*x-2*x^2-x^3)/(x-1)^5 + O(x^50)) \\ _Michel Marcus_, Aug 23 2015
%o (PARI) a(n) = (1/4)*n^2*(n^2+3); \\ _Altug Alkan_, Apr 16 2016
%o (Magma) [n^2*(n^2+3)/4 : n in [1..50]]; // _Wesley Ivan Hurt_, Dec 26 2016
%Y Cf. A000217, A002724, A005353, A052271, A052272, A058001-A058004.
%Y Row n=2 of A246106.
%K nonn,easy,nice
%O 1,2
%A Christian Meland (christian.meland(AT)pfi.no)
%E More terms from _Sam Alexander_
%E Simplified the definition. - _N. J. A. Sloane_, Apr 20 2016
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