A157454 -id:A157454 - cp's OEIS frontend

A204021 Triangle read by rows: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(2i-1,2j-1) (A157454).

1, 1, -1, 2, -4, 1, 4, -12, 9, -1, 8, -32, 40, -16, 1, 16, -80, 140, -100, 25, -1, 32, -192, 432, -448, 210, -36, 1, 64, -448, 1232, -1680, 1176, -392, 49, -1, 128, -1024, 3328, -5632, 5280, -2688, 672, -64, 1, 256, -2304, 8640, -17472, 20592

Offset: 0

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.
a(0)=1 by convention. - Philippe Deléham, Nov 17 2013
The n roots of the n-th polynomial are 1/(1+cos((2*k-1)*Pi/(2*n))) for k = 1..n. See my pdf in the link section for the proof. - Jianing Song, Dec 01 2023

			Top of the triangle:
  1
  1....-1
  2....-4.....1
  4....-12....9....-1
  8....-32....40...-16....1
  16...-80....140..-100...25....-1
  32...-192...432..-448...210...-36....1
  ...
-448=2*(-100)-2*140-(-32). - _Philippe Deléham_, Nov 17 2013

(For references regarding interlacing roots, see A202605.)

Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials
Jianing Song, Roots of the characteristic polynomials

Cf. A157454, A202605, A204016. A085478, A211957.

f[i_, j_] := Min[2 i - 1, 2 j - 1];
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}]]   (* A157454 *)
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[%]                  (* A204021 *)
TableForm[Table[c[n], {n, 1, 10}]]

From Peter Bala, May 01 2012: (Start)
The triangle appears to be a signed version of the row reverse of A211957.
If true, then for 0 <= k <= n-1, T(n,k) = (-1)^k*n/(n-k)*2^(n-k-1)*binomial(2*n-k-1,k) and Sum_{k = 0..n} T(n,k)*x^(n-k) = 1/2*(-1)^n*(b(2*n,-2*x) + 1)/b(n,-2*x), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478.
Conjectural o.g.f.: t*(1-x-x^2*t)/(1-2*t*(1-x)+t^2*x^2) = (1-x)*t + (2-4*x+x^2)*t^2 + .... (End)
T(n,k)=2*T(n-1,k)-2*T(n-1,k-1)-T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 of k<0 or if k>n. - Philippe Deléham, Nov 17 2013

A157455 Number generated by regarding the numbers in row n of A157454 as digits of a base n number.

Original entry on oeis.org

1, 3, 19, 125, 1141, 12943, 182743, 3080025, 60530761, 1357997531, 34237329611, 957927505717, 29446189175677, 986272776455415, 35746439978786671, 1393753996031259953, 58165330912030118161, 2586788074128361802419

Offset: 1

Roger L. Bagula, Mar 01 2009

Note that for odd n, the digits will include n itself.

			Row 4 of A157454 is 1,3,3,1. 1331 in base 4 = 4^3 + 3*4^2 + 3*4 + 1 = 125, so a(4) = 125.

A139038, A003983

t[n_, m_] =Min[1 + 2*m, 1 + 2*(n - m)];
Table[FromDigits[{Table[t[n, m], {m, 0, n}], n + 1}, n + 1], {n, 0, 20}]

t(n,m)=Min[2*m - 1, 2*(n - m) + 1]; a(n)=FromDigits[{Table[t[n, m], {m, 1, n}], n}, n].

Edited by Franklin T. Adams-Watters, Sep 25 2011

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}]]

A098353 Multiplication table of the odd numbers read by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 9, 5, 7, 15, 15, 7, 9, 21, 25, 21, 9, 11, 27, 35, 35, 27, 11, 13, 33, 45, 49, 45, 33, 13, 15, 39, 55, 63, 63, 55, 39, 15, 17, 45, 65, 77, 81, 77, 65, 45, 17, 19, 51, 75, 91, 99, 99, 91, 75, 51, 19, 21, 57, 85, 105, 117, 121, 117, 105, 85, 57, 21

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

a(n) is also the first row of the denominators of the Gram Matrix generated from the normal equations with inner product of the 2D integral with both ranges -1 to 1 over all even 2D polynomials. Subsequent rows and remaining Gram Matrix rows for other 2D polynomials do not currently appear in the OEIS. - John Spitzer, Feb 13 2020

			Array begins:
   1,  3,  5,  7,  9,  11 ...
   3,  9, 15, 21, 27,  33 ...
   5, 15, 25, 35, 45,  55 ...
   7, 21, 35, 49, 63,  77 ...
   9, 27, 45, 63, 81,  99 ...
  11, 33, 55, 77, 99, 121 ...

G. C. Greubel, Antidiagonals n = 1..100

Cf. A003991, A098352, A204022, A157454.

Flat(List([1..12], n-> List([1..n], k-> Maximum(2*k-1, 2*(n-k)+1) *Minimum(2*k-1, 2*(n-k)+1) ))) # G. C. Greubel, Jul 23 2019

[[Max(2*k-1, 2*(n-k)+1)*Min(2*k-1, 2*(n-k)+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 23 2019

seq(seq(max(2*k-1, 2*(n-k)+1)*min(2*k-1, 2*(n-k)+1), k = 1..n), n = 1..12); # G. C. Greubel, Aug 16 2019

Table[Max[2*k-1, 2*(n-k)+1]*Min[2*k-1, 2*(n-k)+1], {n,0,12}, {k,0,n} ]//Flatten (* G. C. Greubel, Jul 23 2019 *)

{T(n, k) = max(2*k-1, 2*(n-k)+1)*min(2*k-1, 2*(n-k)+1)};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 23 2019

[[max(2*k-1, 2*(n-k)+1)*min(2*k-1, 2*(n-k)+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 23 2019

a(n) = A204022(n,k) * A157454(n,k). - Ridouane Oudra, Jul 20 2019

A347026 Irregular triangle read by rows in which row n lists the first n odd numbers, followed by the first n odd numbers in decreasing order.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 7, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 17, 15, 13, 11, 9, 7, 5, 3, 1

Offset: 1

Eddie Gutierrez, Aug 11 2021

The terms of this sequence are the numbers in an irregular triangle corresponding to the addition of rows when multiplying two large numbers via a novel method (see Links).
Sums of the rising diagonals yield sequence A007980.
When the 2n terms in row n are used as the coefficients of a (2n-1)st-order polynomial in x, dividing that polynomial by x+1 produces a (2n-2)nd-order polynomial whose coefficients are the n-th row of A004737 (if that sequence is taken as an irregular triangle with 2n-1 terms in its n-th row). E.g., for n=3, (x^5 + 3x^4 + 5x^3 + 5x^2 + 3x + 1)/(x+1) = x^4 + 2x^3 + 3x^2 + 2x + 1.

			Triangle begins:
  1,  1;
  1,  3,  3,  1;
  1,  3,  5,  5,  3,  1;
  1,  3,  5,  7,  7,  5,  3,  1;
  1,  3,  5,  7,  9,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 11,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 13, 13, 11,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 13, 15, 15, 13, 11,  9,  7,  5,  3,  1;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eddie Gutierrez, A Novel Method for Multiplying Large Numbers.
Eddie Gutierrez, The Column Addition Triangle (CAT)-A Pascal Analog (Part I).
Eddie Gutierrez, The Column Addition Triangle (CAT) and Polynomials.

Even-indexed rows of A157454.
Antidiagonal sums give A007980.
Row lengths give nonzero terms of A005843.
Cf. A004737.

#include 
int main()
{
   int n, k;
   for (n=1; n<=13; n++)
   {
      for (k=1; k<=n; k++)
      {
         printf("%d ", 2*k - 1);
      }
      for (k=n+1; k<=2*n; k++)
      {
         printf("%d ", 4*n - 2*k + 1);
      }
      printf("\n");
   }
   return 0;
}

Array[Join[#, Reverse[#]] &@Range[1, 2 # - 1, 2] &, 9] // Flatten (* Michael De Vlieger, Aug 18 2021 *)
Flatten[Table[Join[Range[1,2n+1,2],Range[2n+1,1,-2]],{n,0,10}]] (* Harvey P. Dale, Aug 31 2024 *)

row(n) = n*=2; vector(n, k, min(2*k-1, 2*(n-k)+1)); \\ Michel Marcus, Aug 17 2021

T(n,k) = 2k - 1 for 1 <= k <= n,

4n - 2k + 1 for n+1 <= k <= 2n.

Better definition from Omar E. Pol, Aug 14 2021

cp's OEIS Frontend

A204021 Triangle read by rows: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(2i-1,2j-1) (A157454).

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A157455 Number generated by regarding the numbers in row n of A157454 as digits of a base n number.

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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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A098353 Multiplication table of the odd numbers read by antidiagonals.

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A347026 Irregular triangle read by rows in which row n lists the first n odd numbers, followed by the first n odd numbers in decreasing order.

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