A106314 -id:A106314 - cp's OEIS frontend

A204020 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i,j)^2 (A106314).

1, -1, 3, -5, 1, 15, -31, 14, -1, 105, -247, 157, -30, 1, 945, -2433, 1892, -553, 55, -1, 10395, -28653, 25573, -9620, 1554, -91, 1, 135135, -393279, 388810, -173773, 37550, -3738, 140, -1, 2027025, -6169455

Offset: 1

Clark Kimberling, Jan 11 2012

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

Constant term of p(n,x) is A001147(n), and the coefficient of the linear term is A000330(n). - Enrique Pérez Herrero, Feb 20 2013

			Top of the array:
1.....-1
3.....-5.....1
15....-31....14....-1
105...-247...157...-30...1

(For references regarding interlacing roots, see A202605.)

Cf. A106314, A202605, A204016.

f[i_, j_] := Min[i^2, j^2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]   (* A106314 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                  (* A204020 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 2, 0, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 0, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 0, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 7

Offset: 1

Clark Kimberling, Jan 10 2012

A204016 represents the matrix M given by f(i,j) = max{(j mod i), (i mod j)} for i >= 1 and j >= 1.  See A204017 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
Guide to symmetric matrices M based on functions f(i,j) and characteristic polynomial sequences (c.p.s.) with interlaced zeros:
f(i,j)..........................M.........c.p.s.
C(i+j,j)........................A007318...A045912
min(i,j)........................A003983...A202672
max(i,j)........................A051125...A203989
(i+j)*min(i,j)..................A203990...A203991
|i-j|...........................A049581...A203993
max(i-j+1,j-i+1)................A143182...A203992
min(i-j+1,j-i+1)................A203994...A203995
min(i(j+1),j(i+1))..............A203996...A203997
max(i(j+1)-1,j(i+1)-1)..........A203998...A203999
min(i(j+1)-1,j(i+1)-1)..........A204000...A204001
min(2i+j,i+2j)..................A204002...A204003
max(2i+j-2,i+2j-2)..............A204004...A204005
min(2i+j-2,i+2j-2)..............A204006...A204007
max(3i+j-3,i+3j-3)..............A204008...A204011
min(3i+j-3,i+3j-3)..............A204012...A204013
min(3i-2,3j-2)..................A204028...A204029
1+min(j mod i, i mod j).........A204014...A204015
max(j mod i, i mod j)...........A204016...A204017
1+max(j mod i, i mod j).........A204018...A204019
min(i^2,j^2)....................A106314...A204020
min(2i-1, 2j-1).................A157454...A204021
max(2i-1, 2j-1).................A204022...A204023
min(i(i+1)/2,j(j+1)/2)..........A106255...A204024
gcd(i,j)........................A003989...A204025
gcd(i+1,j+1)....................A204030...A204111
min(F(i+1),F(j+1)),F=A000045....A204026...A204027
gcd(F(i+1),F(j+1)),F=A000045....A204112...A204113
gcd(L(i),L(j)),L=A000032........A204114...A204115
gcd(2^i-1,2^j-2)................A204116...A204117
gcd(prime(i),prime(j))..........A204118...A204119
gcd(prime(i+1),prime(j+1))......A204120...A204121
gcd(2^(i-1),2^(j-1))............A144464...A204122
max(floor(i/j),floor(j/i))......A204123...A204124
min(ceiling(i/j),ceiling(j/i))..A204143...A204144
Delannoy matrix.................A008288...A204135
max(2i-j,2j-i)..................A204154...A204155
-1+max(3i-j,3j-i)...............A204156...A204157
max(3i-2j,3j-2i)................A204158...A204159
floor((i+1)/2)..................A204164...A204165
ceiling((i+1)/2)................A204166...A204167
i+j.............................A003057...A204168
i+j-1...........................A002024...A204169
i*j.............................A003991...A204170
..abbreviation below:  AOE means "all 1's except"
AOE f(i,i)=i....................A204125...A204126
AOE f(i,i)=A000045(i+1).........A204127...A204128
AOE f(i,i)=A000032(i)...........A204129...A204130
AOE f(i,i)=2i-1.................A204131...A204132
AOE f(i,i)=2^(i-1)..............A204133...A204134
AOE f(i,i)=3i-2.................A204160...A204161
AOE f(i,i)=floor((i+1)/2).......A204162...A204163
...
Other pairs (M, c.p.s.): (A204171, A204172) to (A204183, A204184)
See A202695 for a guide to choices of symmetric matrix M for which the zeros of the characteristic polynomials are all positive.

			Northwest corner:
  0 1 1 1 1 1 1 1
  0 1 2 2 2 2 2 2
  1 2 0 3 3 3 3 3
  1 2 3 0 4 4 4 4
  1 2 3 4 0 5 5 5
  1 2 3 4 5 0 6 6
  1 2 3 4 5 6 0 7

Cf. A204017, A202453.

f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A204016 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]               (* A204017 *)
TableForm[Table[c[n], {n, 1, 10}]]

A168281 Triangle T(n,m) = 2*(min(n - m + 1, m))^2 read by rows.

Original entry on oeis.org

2, 2, 2, 2, 8, 2, 2, 8, 8, 2, 2, 8, 18, 8, 2, 2, 8, 18, 18, 8, 2, 2, 8, 18, 32, 18, 8, 2, 2, 8, 18, 32, 32, 18, 8, 2, 2, 8, 18, 32, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 72, 50, 32, 18, 8, 2, 2, 8, 18, 32, 50, 72, 98, 72

Offset: 1

Paul Curtz, Nov 22 2009

Row sums are A099956(n-1) = 2*A005993(n-1).
The flattened triangle is simply 2 followed by A137508.
If A106314 is interpreted as a triangle, T(n,m) = 2*A106314(n,m).

			The table starts in row n=1 with columns 1<=m<=n as:
  2;
  2,2;
  2,8,2;
  2,8,8,2;
  2,8,18,8,2;
  2,8,18,18,8,2;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10011 (First 141 rows).

Cf. A005993, A099956, A137508, A106314.

A168281 := proc(n,m) 2*(min(n+1-m,m))^2 ; end proc:
seq(seq(A168281(n,m),m=1..n),n=1..20) ;

Table[Map[2 Min[n + # - 1, #]^2 &, Drop[#, -Boole@ EvenQ@ n] ~Join~ Reverse@ # &@ Range@ Floor[n/2]], {n, 2, 14}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)

Rephrased all comments in terms of a triangle by R. J. Mathar, Nov 24 2010
More terms from Michael De Vlieger, Jul 19 2016
Definition corrected by Georg Fischer, Nov 11 2021

A137508 Successive structures of alkaline earth metals (periodic table elements from 2nd column).

Original entry on oeis.org

2, 2, 2, 8, 2, 2, 8, 8, 2, 2, 8, 18, 8, 2, 2, 8, 18, 18, 8, 2, 2, 8, 18, 32, 18, 8, 2

Offset: 1

Paul Curtz, Apr 23 2008

nonn

Apparently a(n) = A168281(n+1). - Georg Fischer, Nov 11 2021

			27 terms: 2, 2 for beryllium, ... Every structure is palindromic (even and odd mixed). Also 2*A106314.

Cf. A005993, A099956, A168281. Same numbers as in A093907.

A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)(n + 1) - ij.

Original entry on oeis.org

1, 1, 5, 72, 2309, 140400, 14495641, 2347782144, 562385930985, 190398813728000, 87889475202276461, 53726132414026874880, 42454821207656237294381, 42495322215073539046387712, 52954624815227996007075890625, 80932107560443542398970529579008, 149736953621087625813286348913927569

Offset: 0

Stefano Spezia, Apr 29 2023

nonn

M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
The elements sum of the matrix M(n) is A002415(n+1).
The antidiagonal sum of the matrix M(n) is A005993(n-1).
The n-th row of A107985 gives the row or column sums of the matrix M(n+1).

			a(3) = 72:
           [3, 2, 1]
    M(3) = [2, 4, 2]
           [1, 2, 3]
a(5) = 140400:
           [5, 4, 3, 2, 1]
           [4, 8, 6, 4, 2]
    M(5) = [3, 6, 9, 6, 3]
           [2, 4, 6, 8, 4]
           [1, 2, 3, 4, 5]

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
Wikipedia, Special unitary group

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
    Matrix(n, (i, j)-> min(i, j)*(n+1)-i*j))):
seq(a(n), n=0..16);  # Alois P. Heinz, Apr 30 2023

M[i_, j_, n_]:=Min[i, j](n+1)-i j; Join[{1}, Table[Permanent[Table[M[i, j, n], {i, n}, {j, n}]], {n, 17}]]

a(n) = matpermanent(matrix(n, n, i, j, min(i, j)*(n + 1) - i*j)); \\ Michel Marcus, Apr 30 2023

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023

A362711 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = min(i, j)(2n + 1) - i*j.

Original entry on oeis.org

1, 1, 17, 1177, 210249, 76961257, 50203153993, 53127675356625, 85252003916011889, 197131843368693693937, 631233222450168374457057

Offset: 0

Stefano Spezia, Apr 30 2023

M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
The elements sum of the matrix M(n) is A002415(n+1).
The antidiagonal sum of the matrix M(n) is A005993(n-1).
The n-th row of A107985 gives the row or column sums of the matrix M(n+1).

			a(2) = 17:
    [4, 3, 2, 1]
    [3, 6, 4, 2]
    [2, 4, 6, 3]
    [1, 2, 3, 4]

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
Wikipedia, Hafnian
Wikipedia, Special unitary group
Wikipedia, Symmetric matrix

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985, A362679 (permanent).

M[i_, j_, n_]:=Part[Part[Table[Min[r,c](n+1)-r c, {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, min(i, j)*(n + 1) - i*j);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

cp's OEIS Frontend

A204020 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i,j)^2 (A106314).

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A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.

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A168281 Triangle T(n,m) = 2*(min(n - m + 1, m))^2 read by rows.

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A137508 Successive structures of alkaline earth metals (periodic table elements from 2nd column).

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A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)*(n + 1) - i*j.

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A362711 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = min(i, j)*(2*n + 1) - i*j.

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A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)(n + 1) - ij.

A362711 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = min(i, j)(2n + 1) - i*j.