A355175 -id:A355175 - cp's OEIS frontend

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

Benoit Cloitre, Feb 01 2003

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

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).

Cf. A001263, A002415, A355175.

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

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

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

A355326 Determinant of the n X n matrix [(i-j)^3+d(i,j)]_{1<=i,j<=n}, where d(i,j) is 1 or 0 according as i = j or not.

Original entry on oeis.org

1, 2, 67, 2157, 96471, 2312410, 32099453, 302049265, 2134677349, 12111035146, 57724828943, 238763085133, 877863236043, 2922096754578, 8932649551321, 25364746314689, 67523106652585, 169800639240178, 405912148130875, 927335183703821, 2033820866612767, 4298718682928682, 8785487346560277, 17412229912018801, 33551232473687501

Offset: 1

Zhi-Wei Sun, Jun 28 2022

nonn

Conjecture 1: a(n) = 1 + P(n^2)*n^2*(n^2-1)/672000, where P(n) = n^6 - 19*n^5 + 123*n^4 - 337*n^3 + 12376*n^2 - 44144*n + 40000.
Conjecture 2: For any positive integers m and n, the determinant of the matrix [(i-j)^m+d(i,j)]_{1<=i,j<=n} has the form 1 + n^2*(n^2-1)*P(n), where P(n) is a polynomial in n with rational number coefficients whose degree is (m+1)^2-4.
See also A079034 and A355175 for related determinants.

			a(3) = 67 since the matrix [(i-j)^3+d(i,j)]_{1<=i,j<=3} = [1,-1,-8;1,1,-1;8,1,1] has determinant 67.

Han Wang and Zhi-Wei Sun, Evaluations of three determinants, arXiv:2206.12317 [math.NT], 2022.

Cf. A000578, A079034, A355175.

a[n_]:=a[n]=Det[Table[If[i==j,1,(i-j)^3],{i,1,n},{j,1,n}]];
Table[a[n],{n,1,25}]

a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, (i-j)^3))); \\ Michel Marcus, Jun 29 2022

from sympy import Matrix
def A355326(n): return Matrix(n,n,[1 if i==j else (i-j)**3 for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Jun 29 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions