A204021 -id:A204021 - cp's OEIS frontend

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

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}]]

A211955 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 4, 1, 10, 30, 28, 8, 1, 15, 70, 112, 72, 16, 1, 21, 140, 336, 360, 176, 32, 1, 28, 252, 840, 1320, 1056, 416, 64, 1, 36, 420, 1848, 3960, 4576, 2912, 960, 128, 1, 45, 660, 3696, 10296, 16016, 14560, 7680, 2176, 256

Offset: 0

Peter Bala, Apr 30 2012

Let b(n,x) = Sum_{k = 0..n} binomial(n+k,2*k)*x^k denote the Morgan-Voyce polynomials of A085478. This triangle lists the coefficients (in ascending powers of x) of the related polynomial sequence R(n,x) := (1/2)*b(n,2*x) + 1/2. Several sequences already in the database are of the form (R(n,x))n>=0 for a fixed value of x. These include A101265 (x = 1), A011900 (x = 2), A182432 (x = 3), A054318 (x = 4) as well as signed versions of A133872 (x = -1), A109613(x = -2), A146983 (x = -3) and A084159 (x = -4).

The polynomials R(n,x) factorize in the ring Z[x] as R(n,x) = P(n,x)*P(n+1,x) for n >= 1: explicitly, P(2*n,x) = 1/2*(b(2*n,2*x) + 1)/b(n,2*x) and P(2*n+1,x) = b(n,2*x). The coefficients of P(n,x) occur in several tables in the database, although without the connection to the Morgan-Voyce polynomials being noted - see A211956 for more details. In terms of T(n,x), the Chebyshev polynomials of the first kind, we have P(2*n,x) = T(2*n,u) and P(2*n+1,x) = 1/u * T(2*n+1,u), where u = sqrt((x+2)/2). Hence R(n,x) = 1/u * T(n,u) * T(n+1,u).

			Triangle begins
.n\k.|..0....1....2....3....4....5....6
= = = = = = = = = = = = = = = = = = = =
..0..|..1
..1..|..1....1
..2..|..1....3....2
..3..|..1....6...10....4
..4..|..1...10...30...28....8
..5..|..1...15...70..112...72...16
..6..|..1...21..140..336..360..176...32

Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

Cf. A011900, A084159, A085478, A101265 (row sums), A109613, A112373, A123519, A133872 (alt row sums), A146983, A182432, A204021, A208513, A211956, A211957.

T(n,0) = 1; T(n,k) = 2^(k-1)*binomial(n+k,2*k) for k > 0.
O.g.f. for column k (except column 0): 2^(k-1)*x^k/(1-x)^(2*k+1).
O.g.f.: (1-t*(x+2)+t^2)/((1-t)*(1-2*t(x+1)+t^2)) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2 + ....
Removing the first column from the triangle produces the Riordan array (x/(1-x)^3, 2*x/(1-x)^2).
The row polynomials R(n,x) := 1/2*b(n,2*x) + 1/2 = 1 + x*Sum_{k = 1..n} binomial(n+k,2*k)*(2*x)^(k-1).
Recurrence equation: R(n,x) = 2*(1+x)*R(n-1,x) - R(n-2,x) - x with initial conditions R(0,x) = 1, R(1,x) = 1+x.
Another recurrence is R(n,x)*R(n-2,x) = R(n-1,x)*(R(n-1,x) + x).
With P(n,x) as defined in the Comments section we have (x+2)/x - {Sum_{k = 0..2n} 1/R(k,x)}^2 = 2/(x*P(2*n+1,x)^2); (x+2)/x - {Sum_{k = 0..2n+1} 1/R(k,x)}^2 = (x+2)/(x*P(2*n+2,x)^2); consequently Sum_{k >= 0} 1/R(k,x) = sqrt((x+2)/x) for either x > 0 or x <= -2.
Row sums R(n,1) = A101265(n+1); Alt. row sums R(n,-1) = A133872(n+1);
R(n,2) = A011900(n); R(n,-2) = (-1)^n * A109613(n); R(n,3) = A182432;
R(n,-3) = (-1)^n * A146983(n); R(n,4) = A054318(n+1); R(n,-4) = (-1)^n * A084159(n).

A211956 Coefficients of a sequence of polynomials related to the Morgan-Voyce polynomials.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 4, 2, 1, 6, 4, 1, 9, 12, 4, 1, 12, 20, 8, 1, 16, 40, 32, 8, 1, 20, 60, 56, 16, 1, 25, 100, 140, 80, 16, 1, 30, 140, 224, 144, 32, 1, 36, 210, 448, 432, 192, 32, 1, 42, 280, 672, 720, 352, 64, 1, 49, 392, 1176, 1680, 1232, 448, 64

Offset: 0

Peter Bala, Apr 30 2012

The row generating polynomials R(n,x) of A211955 factorize in the ring Z[x] as R(n,x) = P(n,x)*P(n+1,x) for n >= 1: explicitly, P(2*n,x) = 1/2*(b(2*n,2*x) + 1)/b(n,2*x) and P(2*n+1,x) = 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. This triangle lists the coefficients in ascending powers of x of the polynomials P(n,x).

The odd numbered rows of the present triangle produce triangle A123519; the even numbered row entries are recorded separately in A211957 and appear to equal the unsigned and row reversed form of A204021. The even numbered rows with a factor of 2^(k-1) removed from the k-th column entries produce triangle A208513.

			Triangle begins
.n\k.|..0....1....2....3....4
= = = = = = = = = = = = = = =
..0..|..1
..1..|..1
..2..|..1....1
..3..|..1....2
..4..|..1....4....2
..5..|..1....6....4
..6..|..1....9...12....4
..7..|..1...12...20....8
..8..|..1...16...40...32....8
..9..|..1...20...60...56...16
...

Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

Cf. A005246 (row sums), A085478, A123519, A204021, A208513, A211955, A211957.

T(n,0) = 1; for k > 0, T(2*n,k) = 2^k * binomial(n+k,2*k) = A123519(n,k);
for k > 0, T(2*n-1,k) = n/(n+k)*(2^k)*binomial(n+k,2*k) = 2^(k-1)*A208513(n,k).
O.g.f.: ((1+t)*(1-t^2)-t^2*x)/((1-t^2)^2-2*t^2*x) = 1 + t + (1+x)*t^2 + (1+2*x)*t^3 + (1+4*x+2*x^2)*t^4 + ....
Row generating polynomials: P(2*n,x) := 1/2*(b(2*n,2*x)+1)/b(n,2*x) and P(2*n+1,x) := 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.
The product P(n,x)*P(n+1,x) is the n-th row polynomial of A211955.
In terms of T(n,x), the Chebyshev polynomials of the first kind, we have P(2*n,x) = T(2*n,u) and P(2*n+1,x) = 1/u*T(2*n+1,u), where u = sqrt((x+2)/2).
Other representations for the row polynomials include
P(2*n,x) = 1/2*(1+x+sqrt(x^2+2*x))^n + 1/2*(1+x-sqrt(x^2+2*x))^n;
P(2*n,x) = n*Sum_{k = 0..n}(-1)^(n-k)/(n+k) * binomial(n+k,2*k) * (2*x+4)^k for n >= 1;
P(2*n+1,x) = (2*n+1)*Sum_{k=0..n} (-1)^(n-k)/(n+k+1) * binomial(n+k+1,2*k+1) * (2*x+4)^k.
Recurrence equation: P(n+1,x)*P(n-2,x) - P(n,x)*P(n-1,x) = x.
Row sums A005246(n+2).

A211957 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 12, 4, 1, 16, 40, 32, 8, 1, 25, 100, 140, 80, 16, 1, 36, 210, 448, 432, 192, 32, 1, 49, 392, 1176, 1680, 1232, 448, 64, 1, 64, 672, 2688, 5280, 5632, 3328, 1024, 128, 1, 81, 1080, 5544, 14256, 20592, 17472, 8640, 2304, 256, 1, 100, 1650, 10560, 34320, 64064, 72800, 51200, 21760, 5120, 512

Offset: 0

Peter Bala, Apr 30 2012

Triangle formed from the even numbered rows of A211956.
The coefficients of the  Morgan-Voyce polynomials b(n,x) := sum {k = 0..n} binomial(n+k,2*k)*x^k are listed in A085478. The rational functions 1/2*(b(2*n,2*x) + 1)/b(n,2*x) turn out to be integer polynomials. Their coefficients are listed in this triangle. These polynomials occur as factors of the row polynomials R(n,x) of A211955.
This triangle appears to be the row reverse of the unsigned triangle |A204021|.

			Triangle begins
.n\k.|..0....1....2....3....4....5....6....7
= = = = = = = = = = = = = = = = = = = = = = =
..0..|..1
..1..|..1....1
..2..|..1....4....2
..3..|..1....9...12....4
..4..|..1...16...40...32....8
..5..|..1...25..100..140...80...16
..6..|..1...36..210..448..432..192...32
..7..|..1...49..392.1176.1680.1232..448...64

R. J. Mathar, Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice, vixra:2208.0111
Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

A085478, A111125, A204021, A211955, A211956.

T(n,0) = 1 and for k > 0, T(n,k) = n/k*2^(k-1)*binomial(n+k-1,2*k-1) = 2^(k-1)*A208513(n,k).
O.g.f.: ((1-t)-t*x)/((1-t)^2-2*t*x) = 1 + (1+x)*t + (1+4*x+2*x^2)*t^2 + ....
n-th row polynomial R(n,x) = 1/2*(b(2*n,2*x) + 1)/b(n,2*x) = T(2*n,u), where u = sqrt((x+2)/2) and T(n,u) denotes the Chebyshev polynomial of the first kind.
T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 16 2013

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A211955 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A211956 Coefficients of a sequence of polynomials related to the Morgan-Voyce polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A211957 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula