OFFSET
1,2
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
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.
FORMULA
a(n) = (-1)^(n-1)*(n+1)*(2*n+1)*n^(n-2)*((n+2)^n - n^n)/12.
EXAMPLE
a(4) = -62400 = determinant
| 1, 4, 9, 16|
|16, 1, 4, 9|
| 9, 16, 1, 4|
| 4, 9, 16, 1|
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010
STATUS
approved