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%I #27 Jun 28 2022 10:59:14
%S 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
%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 <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*(x^4-4*x^3+7*x^2-3*x+1) / (x-1)^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 1,2
%A _Benoit Cloitre_, Feb 01 2003
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