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Determinant of M(n), the n X n matrix defined by m(i,i) = 1, m(i,j) = i-j.
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%I #37 Oct 23 2024 12:45:49

%S 1,1,2,7,21,51,106,197,337,541,826,1211,1717,2367,3186,4201,5441,6937,

%T 8722,10831,13301,16171,19482,23277,27601,32501,38026,44227,51157,

%U 58871,67426,76881,87297,98737,111266,124951,139861,156067,173642,192661,213201,235341

%N Determinant of M(n), the n X n matrix defined by m(i,i) = 1, m(i,j) = i-j.

%C Starting (1, 1, 2, 7, 21, 51, 106, ...), = Narayana transform (A001263) of [1, 0, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Jan 04 2008

%C In 2022, Han Wang and Zhi-Wei Sun provided a proof of the formula a(n) = 1 + n^2*(n^2-1)/12 via eigenvalues. See A355175 for my conjecture on det[(i-j)^2+d(i,j)]_{1<=i,j<=n}, where d(i,j) is 1 or 0 according as i = j or not. - _Zhi-Wei Sun_, Jun 28 2022

%H Han Wang and Zhi-Wei Sun, <a href="https://arxiv.org/abs/2206.12317">Evaluations of three determinants</a>, arXiv:2206.12317 [math.NT], 2022.

%H Han Wang and Zhi-Wei Sun, <a href="https://doi.org/10.1017/S000497272400039X">Characteristic polynomials of the matrices with (j, k)-entry q^(j±k) + t</a>, Bull. Australian Math. Soc. (2024). See references.

%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^4-n^2+12)/12; a(n) = A002415(n)+1.

%F G.f.: (x^4-3*x^3+7*x^2-4*x+1) / (1-x)^5. - _Colin Barker_, Jun 24 2013

%t LinearRecurrence[{5,-10,10,-5,1},{1,2,7,21,51},50] (* _Harvey P. Dale_, Aug 17 2014 *)

%Y Cf. A001263, A002415, A355175.

%K nonn,easy

%O 0,3

%A _Benoit Cloitre_, Feb 01 2003

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 23 2024