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

1, -15, 686, -62400, 9406375, -2117816064, 665460727820, -278158506983424, 149228699913026685, -99940926131200000000, 81720620766038589115418, -80119979953874981093376000, 92770427931597143858070722691, -125252587064115948721297529241600

Offset: 1

Author

Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010

Keywords

• easy
• sign

Comments

This sequence is the solution to Problem #11467, proposed by Xiang Qian Chang, in the December 2009 issue of the American Mathematical Monthly.

Please notice that "The Wohascum County Problem Book" predates the Problem #11467 by 16 years. - Robert G. Wilson v, Aug 31 2014

Examples

			a(4) = -62400 = determinant
  | 1,  4,  9, 16|
  |16,  1,  4,  9|
  | 9, 16,  1,  4|
  | 4,  9, 16,  1|
		

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..200
• Xiang Qian Chang, Problem 11467, The American Mathematical Monthly, Vol. 116, No. 10 (Dec., 2009), p. 940.

Crossrefs

Cf. A118707.

Programs

Magma

[(-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

Mathematica

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 *)

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

SageMath

[(-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

Formula

a(n) = (-1)^(n-1)*(n+1)*(2*n+1)*n^(n-2)*((n+2)^n - n^n)/12.