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A177230 a(n) = determinant of n X n circulant matrix whose first row consists of the first n squares (beginning with 1).

%I A177230 #16 Apr 13 2024 01:50:05
%S A177230 1,-15,686,-62400,9406375,-2117816064,665460727820,-278158506983424,
%T A177230 149228699913026685,-99940926131200000000,81720620766038589115418,
%U A177230 -80119979953874981093376000,92770427931597143858070722691,-125252587064115948721297529241600
%N A177230 a(n) = determinant of n X n circulant matrix whose first row consists of the first n squares (beginning with 1).
%C A177230 This sequence is the solution to Problem #11467, proposed by Xiang Qian Chang, in the December 2009 issue of the American Mathematical Monthly.
%C A177230 Please notice that "The Wohascum County Problem Book" predates the Problem #11467 by 16 years. - _Robert G. Wilson v_, Aug 31 2014
%D A177230 George T. Gilbert, Mark I. Krusemeyer and Loren C. Larson, The Wohascum County Problem Book, The Mathematical Association of America, Dolciani Mathematical Expositions No. 14, 1993, problem 130 "Prove that det(...) = (-1)^(n-1)n^(n-2)(n+1)(2n+1)((n+2)^n-n^n)/12", page 31 and solution on page 216.
%H A177230 G. C. Greubel, <a href="/A177230/b177230.txt">Table of n, a(n) for n = 1..200</a>
%H A177230 Xiang Qian Chang, <a href="http://www.jstor.org/stable/40391256">Problem 11467</a>, The American Mathematical Monthly, Vol. 116, No. 10 (Dec., 2009), p. 940.
%F A177230 a(n) = (-1)^(n-1)*(n+1)*(2*n+1)*n^(n-2)*((n+2)^n - n^n)/12.
%e A177230 a(4) = -62400 = determinant
%e A177230   | 1,  4,  9, 16|
%e A177230   |16,  1,  4,  9|
%e A177230   | 9, 16,  1,  4|
%e A177230   | 4,  9, 16,  1|
%t A177230 a[n_] := (-1)^(n - 1) n^(n - 2) (n + 1) (2 n + 1) ((n + 2)^n - n^n)/12; Array[a, 14] (* _Robert G. Wilson v_, Aug 31 2014 *)
%o A177230 (PARI) a(n) = (-1)^(n-1)*(n+1)*(2*n+1)*n^(n-2)*((n+2)^n-n^n)/12 \\ _Charles R Greathouse IV_, Aug 31 2014
%o A177230 (Magma)
%o A177230 [(-1)^n*n^(n-2)*(n^n-(n+2)^n)*Binomial(2*n+2,2)/12: n in [1..30]]; // _G. C. Greubel_, Apr 12 2024
%o A177230 (SageMath)
%o A177230 [(-1)^n*n^(n-2)*(n^n-(n+2)^n)*binomial(2*n+2,2)/12 for n in range(1,31)] # _G. C. Greubel_, Apr 12 2024
%Y A177230 Cf. A118707.
%K A177230 easy,sign
%O A177230 1,2
%A A177230 Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010