A038223 -id:A038223 - cp's OEIS frontend

A223999 T(n,k)=Number of nXk 0..2 arrays with antidiagonals unimodal and rows and diagonals nondecreasing.

3, 6, 9, 10, 31, 27, 15, 76, 157, 81, 21, 155, 476, 793, 243, 28, 281, 1144, 2980, 4004, 729, 36, 469, 2403, 7927, 18672, 20216, 2187, 45, 736, 4614, 17929, 55333, 117386, 102069, 6561, 55, 1101, 8291, 36845, 132119, 388598, 739672, 515338, 19683, 66

Offset: 1

R. H. Hardin Mar 30 2013

Table starts
.....3........6........10........15.........21.........28..........36
.....9.......31........76.......155........281........469.........736
....27......157.......476......1144.......2403.......4614........8291
....81......793......2980......7927......17929......36845.......71061
...243.....4004.....18672.....55333.....132119.....281271......559188
...729....20216....117386....388598.....984595....2160036.....4368458
..2187...102069....739672...2743444....7400832...16795265....34534687
..6561...515338...4664776..19437479...55978489..131782267...276286000
.19683..2601899..29428242.138010718..425257387.1040869367..2229871293
.59049.13136773.185670484.981047716.3240026429.8260503068.18115082917

			Some solutions for n=3 k=4
..0..0..0..1....1..1..2..2....0..0..0..0....2..2..2..2....0..0..0..0
..0..1..2..2....0..1..1..2....0..0..2..2....0..2..2..2....0..0..2..2
..2..2..2..2....0..0..1..2....0..0..1..2....2..2..2..2....2..2..2..2

R. H. Hardin, Table of n, a(n) for n = 1..449

Column 1 is A000244
Column 2 is A038223
Row 1 is A000217(n+1)

Empirical: columns k=1..7 have recurrences of order 1,3,9,18,28,39,54 for n>0,0,0,19,31,44,61

Empirical: rows n=1..7 are polynomials of degree 2*n for k>0,0,2,4,6,8,10

A123218 Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)(2x)^k(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2) binomial(n - floor((k + 1)/2), floor(k/2)).

Original entry on oeis.org

1, 1, -2, -1, 1, -2, -6, 2, 1, 1, -2, -11, 12, 11, -2, -1, 1, -2, -16, 22, 46, -22, -16, 2, 1, 1, -2, -21, 32, 106, -92, -106, 32, 21, -2, -1, 1, -2, -26, 42, 191, -212, -396, 212, 191, -42, -26, 2, 1, 1, -2, -31, 52, 301, -382, -1011, 792, 1011, -382, -301, 52, 31, -2, -1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 04 2006

			Triangle begins with:
1;
1, -2,  -1;
1, -2,  -6,  2,  1;
1, -2, -11, 12, 11,  -2,  -1;
1, -2, -16, 22, 46, -22, -16, 2, 1;

G. C. Greubel, Rows n = 1..101 of triangle, flattened
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A038223, A122600, A122753.

f[n_, k_]:= (-1)^Floor[(k+1)/2]*Binomial[n -Floor[(k+1)/2], Floor[k/2]]; Table[CoefficientList[Sum[f[n, k]*(2*x)^k*(1-x^2)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten

Let f(n, k) = (-1)^floor((k + 1)/2)*binomial(n - floor((k + 1)/2), floor(k/2)) then the polynomials P(n, k, x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k) for an irregular triangle of coefficients.

A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.

Original entry on oeis.org

1, 2, -1, 0, -4, 1, 2, -9, 8, -1, -2, -3, 19, -12, 1, -4, -6, 47, -55, 18, -1, 2, 15, 0, -88, 93, -24, 1, 2, 23, -7, -190, 324, -182, 32, -1, 0, -12, -63, 62, 332, -554, 274, -40, 1, 2, -9, -108, 133, 678, -1642, 1346, -450, 50, -1, -2, -11, 55, 276, -463, -1129, 2832, -2128, 630, -60, 1, -4, -30, 71, 543, -1044, -2204, 7761

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 16 2006

Based on the idea that the Steinbach matrices form a "golden Field". Matrices are: {{2, 2}, {2, 2}}, {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}, {{2, 2, 2, 2}, {2, 3, 3, 2}, {2, 3, 3, 3}, {2, 2, 3, 4}}, {{2, 2, 2, 2, 2}, {2, 3, 3, 3, 2}, {2, 3, 4, 3, 3}, {2, 3, 3, 4, 4}, {2, 2, 3, 4, 5}}, {{2, 2, 2, 2, 2, 2}, {2,3, 3, 3, 3, 2}, {2, 3, 4, 4, 3, 3}, {2, 3, 4, 4, 4, 4}, {2, 3, 3, 4, 5, 5}, {2, 2, 3, 4, 5, 6}}

			{1},
{2, -1},
{0, -4, 1},
{2, -9, 8, -1},
{-2, -3, 19, -12, 1},
{-4, -6,47, -55, 18, -1}
{2, 15, 0, -88, 93, -24, 1},
{2, 23, -7, -190, 324, -182, 32, -1},
{0, -12, -63, 62, 332, -554, 274, -40, 1}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A038223, A038223, A076756, A054142.

An[d_] := Table[Min[n, m] + If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d,1, 20}]]; Flatten[%]

d-th level M(i,j)->An[d]; T(n,m)=CoefficientList[CharacteristicPolynomial[An[d], x], x]

cp's OEIS Frontend

A223999 T(n,k)=Number of nXk 0..2 arrays with antidiagonals unimodal and rows and diagonals nondecreasing.

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A123218 Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)(2x)^k(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2) binomial(n - floor((k + 1)/2), floor(k/2)).

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A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.

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cp's OEIS Frontend

A223999 T(n,k)=Number of nXk 0..2 arrays with antidiagonals unimodal and rows and diagonals nondecreasing.

Views

Author

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Examples

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Crossrefs

Formula

A123218 Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2)* binomial(n - floor((k + 1)/2), floor(k/2)).

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Author

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Crossrefs

Programs

Mathematica

Formula

A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

Formula

A123218 Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)(2x)^k(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2) binomial(n - floor((k + 1)/2), floor(k/2)).