A079034 Determinant of M(n), the n X n matrix defined by m(i,i) = 1, m(i,j) = i-j.

1, 1, 2, 7, 21, 51, 106, 197, 337, 541, 826, 1211, 1717, 2367, 3186, 4201, 5441, 6937, 8722, 10831, 13301, 16171, 19482, 23277, 27601, 32501, 38026, 44227, 51157, 58871, 67426, 76881, 87297, 98737, 111266, 124951, 139861, 156067, 173642, 192661, 213201, 235341

Offset: 0

Author

Benoit Cloitre, Feb 01 2003

Keywords

• nonn
• easy

Comments

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

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

Links

• Han Wang and Zhi-Wei Sun, Evaluations of three determinants, arXiv:2206.12317 [math.NT], 2022.
• Han Wang and Zhi-Wei Sun, Characteristic polynomials of the matrices with (j, k)-entry q^(j±k) + t, Bull. Australian Math. Soc. (2024). See references.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A001263, A002415, A355175.

Programs

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{1,2,7,21,51},50] (* Harvey P. Dale, Aug 17 2014 *)

Formula

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

G.f.: (x^4-3*x^3+7*x^2-4*x+1) / (1-x)^5. - Colin Barker, Jun 24 2013

Extensions

a(0)=1 prepended by Alois P. Heinz, Oct 23 2024