A045912 -id:A045912 - cp's OEIS frontend

A092378 The O(1) loop model on the square lattice is defined as follows: At every vertex the loop turns to the left or to the right with equal probability, unless the vertex has been visited before, in which case the loop leaves the vertex via the unused edge. Every vertex is visited twice. The probability that a face of the lattice on an n X infinity cylinder is surrounded by six loops is conjectured to be given by a(n)/A_{HT}(n)^2, where A_{HT}(n) is the number of n X n half turn symmetric alternating sign matrices.

1, 1, 14061141, 54177740, 659506609478464, 9256643548177084, 155695310201316677915943, 7642657907144601059593232, 220353621720787947087602631723527

Offset: 12

Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 12..60
Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 259-264.

Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092379, A092380, A092381, A092382.

M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
Q[n_?EvenQ, m_]:= Sum[(-1)^r*((m+2*r)/(m+r))*Binomial[m +r, r]*c[n/2-m -2*r, n], {r, 0, (n-2*m)/4}];
Q[n_?OddQ, m_]:= Sum[(-1)^r*Binomial[m+r, r]*(c[(n-1)/2 -m-2*r, n] - c[(n-1)/2 -m-2*r-1, n]), {r, 0, (n-2*m-1)/4}];
Table[Q[n, 6], {n, 12, 30}] (* G. C. Greubel, Nov 15 2019 *)

Even n: Q(n, m) = Sum_{r=0..(n-2*m)/4} (-1)^r * ((m+2*r)/(m+r)) * binomial(m+r, r) * C_{n/2 - m - 2*r}(n).

Odd n: Q(n, m) = Sum_{r=0..(n-2*m-1)/4)} (-1)^r * binomial(m+r,r) * ( C_{(n-1)/2 - m - 2*r}(n) - C_{(n-1)/2 - m - 2*r - 1}(n) ), where the c_{k}(n) are the absolute values of the coefficients of the characteristic polynomial of the n X n Pascal matrix P_{i, j} = binomial(i+j-2, i-1). The sequence is given by Q(n, 6).

A092380 The O(1) loop model on the square lattice is defined as follows: At every vertex the loop turns to the left or to the right with equal probability, unless the vertex has been visited before, in which case the loop leaves the vertex via the unused edge. Every vertex is visited twice. The probability that a face of the lattice on an n X infinity cylinder is surrounded by eight loops is conjectured to be given by a(n)/A_{HT}(n)^2, where A_{HT}(n) is the number of n X n half turn symmetric alternating sign matrices.

Original entry on oeis.org

1, 1, 3143981871, 12219117170, 26773657259138210984, 386199802888523031294, 982474651752126202075575490369, 50748123995890025746709567402256, 191795630733414647568032678703215924098176

Offset: 16

Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 16..60
Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 259-264.
Saibal Mitra and Bernard Nienhuis, Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders, arXiv:cond-mat/0407578 [cond-mat.stat-mech], 2004.
Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, arXiv:math-ph/0312036, 2003.

Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092389, A092381, A092382.

M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
Q[n_?EvenQ, m_]:= Sum[(-1)^r*((m+2*r)/(m+r))*Binomial[m +r, r]*c[n/2 -m-2*r, n], {r, 0, (n-2*m)/4}];
Q[n_?OddQ, m_]:= Sum[(-1)^r*Binomial[m+r, r]*(c[(n-1)/2 -m-2*r, n] - c[(n-1)/2 -m-2*r-1, n]), {r, 0, (n-2*m-1)/4}];
Table[Q[n, 8], {n, 16, 40}] (* G. C. Greubel, Nov 16 2019 *)

Even n: Q(n, m) = Sum_{r=0..(n-2*m)/4} (-1)^r * ((m+2*r)/(m+r)) * binomial(m+r, r) * C_{n/2-m-2*r}(n).

Odd n: Q(n, m) = Sum_{r=0..(n-2*m-1)/4)} (-1)^r * binomial(m+r,r) * ( C_{(n-1)/2 -m-2*r}(n) - C_{(n-1)/2 -m-2*r-1}(n) ), where the c_{k}(n) are the absolute values of the coefficients of the characteristic polynomial of the n X n Pascal matrix P_{i, j} = binomial(i+j-2, i-1). The sequence is given by Q(n, 8).

More terms added by G. C. Greubel, Nov 16 2019

A092381 The O(1) loop model on the square lattice is defined as follows: At every vertex the loop turns to the left or to the right with equal probability, unless the vertex has been visited before, in which case the loop leaves the vertex via the unused edge. Every vertex is visited twice. The probability that a face of the lattice on an n X infinity cylinder is surrounded by nine loops is conjectured to be given by a(n)/A_{HT}(n)^2, where A_{HT}(n) is the number of n X n half turn symmetric alternating sign matrices.

Original entry on oeis.org

1, 1, 47564380971, 185410909790, 5599434135148010392903, 81562945655108319592717, 2647122748975437613370942794822122, 139318635878972598351963980703033608, 6292966726927006717847495753884145618797281792

Offset: 18

Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 18..60
Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 259-264.
Saibal Mitra and Bernard Nienhuis, Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders, arXiv:cond-mat/0407578 [cond-mat.stat-mech], 2004.
Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, arXiv:math-ph/0312036, 2003.

Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092382.

M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
Q[n_?EvenQ, m_]:= c[(n-2*m)/2, n] + Sum[(-1)^r*((m+2*r)/(m+r))*Binomial[m +r, r]*c[n/2 -m-2*r, n], {r, (n-2*m)/4}];
Q[n_?OddQ, m_]:= Sum[(-1)^r*Binomial[m+r, r]*(c[(n-1)/2 -m-2*r, n] - c[(n-1)/2 -m-2*r-1, n]), {r, 0, (n-2*m-1)/4}];
Table[Q[n, 9], {n, 18, 40}] (* G. C. Greubel, Nov 16 2019 *)

Even n: Q(n, m) = Sum_{r=0..(n-2*m)/4} (-1)^r * ((m+2*r)/(m+r)) * binomial(m+r, r) * C_{n/2-m- 2*r}(n).

Odd n: Q(n, m) = Sum_{r=0..(n-2*m-1)/4)} (-1)^r * binomial(m+r,r) * ( C_{(n-1)/2 -m-2*r}(n) - C_{(n-1)/2 -m-2*r-1}(n) ), where the c_{k}(n) are the absolute values of the coefficients of the characteristic polynomial of the n X n Pascal matrix P_{i, j} = binomial(i+j-2, i-1). The sequence is given by Q(n, 9).

More terms added by G. C. Greubel, Nov 16 2019

A092382 The O(1) loop model on the square lattice is defined as follows: At every vertex the loop turns to the left or to the right with equal probability, unless the vertex has been visited before, in which case the loop leaves the vertex via the unused edge. Every vertex is visited twice. The probability that a face of the lattice on an n X infinity cylinder is surrounded by ten loops is conjectured to be given by a(n)/A_{HT}(n)^2, where A_{HT}(n) is the number of n X n half turn symmetric alternating sign matrices.

Original entry on oeis.org

1, 1, 723668784231, 2827767747950, 1193097790725426305663064, 17520037013918467453246138, 7392624504986931437972335103490414473, 395235071756082109802989440265119512888, 218243704050866770455587351635302655565432102527624

Offset: 20

Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 20..70
Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 259-264.
Saibal Mitra and Bernard Nienhuis, Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders, arXiv:cond-mat/0407578 [cond-mat.stat-mech], 2004.
Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, arXiv:math-ph/0312036, 2003.

Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092381.

M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
Q[n_?EvenQ, m_]:= c[(n-2*m)/2, n] + Sum[(-1)^r*((m+2*r)/(m+r))*Binomial[m +r, r]*c[n/2 -m-2*r, n], {r, (n-2*m)/4}];
Q[n_?OddQ, m_]:= Sum[(-1)^r*Binomial[m+r, r]*(c[(n-1)/2 -m-2*r, n] - c[(n-1)/2 -m-2*r-1, n]), {r, 0, (n-2*m-1)/4}];
Table[Q[n, 10], {n, 20, 40}] (* G. C. Greubel, Nov 16 2019 *)

Even n: Q(n, m) = C_{n/2-m}(n) + Sum_{r=1..(n-2*m)/4} (-1)^r * ((m+2*r)/(m+r)) * binomial(m+r, r) * C_{n/2-m- 2*r}(n).

Odd n: Q(n, m) = Sum_{r=0..(n-2*m-1)/4)} (-1)^r * binomial(m+r,r) * ( C_{(n-1)/2 -m-2*r}(n) - C_{(n-1)/2 -m-2*r-1}(n) ), where the c_{k}(n) are the absolute values of the coefficients of the characteristic polynomial of the n X n Pascal matrix P_{i, j} = binomial(i+j-2, i-1). The sequence is given by Q(n, 10).

More terms added by G. C. Greubel, Nov 16 2019

A092379 The O(1) loop model on the square lattice is defined as follows: At every vertex the loop turns to the left or to the right with equal probability, unless the vertex has been visited before, in which case the loop leaves the vertex via the unused edge. Every vertex is visited twice. The probability that a face of the lattice on an n X infinity cylinder is surrounded by seven loops is conjectured to be given by a(n)/A_{HT}(n)^2, where A_{HT}(n) is the number of n X n half turn symmetric alternating sign matrices.

Original entry on oeis.org

1, 1, 209295261, 810375650, 130981854694547781, 1866712378783655407, 380792413068640291929758918, 19226936188283951521093833164, 6245082121880029165837197634771465822, 1084566535537396419423204907970597478243

Offset: 14

Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 14..60
Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, (2003) pp. 259-264.

Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092380, A092381, A092382.

M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
Q[n_?EvenQ, m_]:= Sum[(-1)^r*((m+2*r)/(m+r))*Binomial[m +r, r]*c[n/2 -m-2*r, n], {r, 0, (n-2*m)/4}];
Q[n_?OddQ, m_]:= Sum[(-1)^r*Binomial[m+r, r]*(c[(n-1)/2 -m-2*r, n] - c[(n-1)/2 -m-2*r-1, n]), {r, 0, (n-2*m-1)/4}];
Table[Q[n, 7], {n, 14, 30}] (* Jean-François Alcover, Sep 11 2012; modified by G. C. Greubel, Nov 15 2019 *)

Even n: Q(n, m) = Sum_{r=0..(n-2*m)/4} (-1)^r * ((m+2*r)/(m+r)) * binomial(m+r, r) * C_{n/2 - m - 2*r}(n).

Odd n: Q(n, m) = Sum_{r=0..(n-2*m-1)/4)} (-1)^r * binomial(m+r,r) * ( C_{(n-1)/2 - m - 2*r}(n) - C_{(n-1)/2 - m - 2*r - 1}(n) ), where the c_{k}(n) are the absolute values of the coefficients of the characteristic polynomial of the n X n Pascal matrix P_{i, j} = binomial(i+j-2, i-1). The sequence is given by Q(n, 7).

More terms added and edited by G. C. Greubel, Nov 15 2019

A123948 Triangle read by rows: first row is 1, and n-th row (n > 0) gives the coefficients in the expansion of the characteristic polynomial of the (n - 1)-th Bernstein basis matrix, horizontal flipped.

Original entry on oeis.org

1, 1, -1, -1, 1, 1, -2, 3, 3, -1, 9, -15, -22, 7, 1, 96, -184, -314, 139, 19, -1, -2500, 5250, 10575, -5375, -1026, 51, 1, -162000, 369900, 842310, -498171, -111179, 7644, 141, -1, 26471025, -64790985, -164634169, 109325076, 28870212, -2322404, -59193, 393, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 26 2006

The Bernstein basis matrix of order n - 1 is an n X n matrix whose m-th row represents the coefficients in the expansion of the Bernstein basis polynomial defined as binomial(n, m)*x^m*(1 - x)^(n - m), 0 <= m <= n - 1. For n = 0, we obtain the 0 X 0 matrix. The convention is that the characteristic polynomial of the empty matrix is identically 1 (see [de Boor] and [Johnson et al.]). Row n of the present sequence is obtained by taking the characteristic polynomial of the matrix represented by the polynomials binomial(n, m)*x^(n - m)*(1 - x)^m. The resulting matrix is, in fact, the horizontal flipped version of the Bernstein basis matrix of order n (see example). - Franck Maminirina Ramaharo, Oct 19 2018

			Triangle begins:
        1;
        1,     -1;
       -1,      1,      1;
       -2,      3,      3,      -1;
        9,    -15,    -22,       7,       1;
       96,   -184,   -314,     139,      19,   -1;
    -2500,   5250,  10575,   -5375,   -1026,   51,   1;
  -162000, 369900, 842310, -498171, -111179, 7644, 141, -1;
      ...
From _Franck Maminirina Ramaharo_, Oct 19 2018: (Start)
Let n = 6 (i.e., order 5). The corresponding Bernstein basis matrix has the form
   1, -5,  10, -10,   5,  -1
   0,  5, -20,  30, -20,   5
   0,  0,  10, -30,  30, -10
   0,  0,   0,  10, -20,  10
   0,  0,   0,   0,   5,  -5
   0,  0,   0,   0,   0,   1.
Flipping this matrix horizontally gives the matrix for the polynomials binomial(5, m)*x^(5 - m)*(1 - x)^m, 0 <= m <= 5,
   0,  0,   0,   0,   0,   1
   0,  0,   0,   0,   5,  -5
   0,  0,   0,  10, -20,  10
   0,  0,  10, -30,  30, -10
   0,  5, -20,  30, -20,   5
   1, -5,  10, -10,   5,  -1
whose characteristic polynomial is -2500 + 5250*x + 10575*x^2 - 5375*x^3 - 1026*x^4 + 51*x^5 + x^6. (End)

Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, Chap. 30.

Carl de Boor, An empty exercise
John Burkardt, BERNSTEIN_POLYNOMIAL - The Bernstein Polynomials
Charles R. Johnson and Carlos M. Saiago, Eigenvalues, Multiplicities and Graphs, Cambridge University Press, 2018, p. 8.
Kenneth I. Joy, On-Line Geometric Modeling Notes
Wikipedia, Bernstein polynomial

Cf. A045912, A136678.

M[n_] := Table[CoefficientList[Binomial[n - 1, n - i - 1]*(1 - x)^i*x^(n - i - 1), x], {i, 0, n - 1}];
Join[{1}, Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10}]]//Flatten

Edited, new name, offset corrected by Franck Maminirina Ramaharo, Oct 19 2018

A168228 Coefficient triangle sequence of characteristic polynomials of a Fermat like matrix:M(n)=Pascal n-th matrix: F(n)=Inverse[Transpose[M(n)]].M(n).

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 0, 1, 1, 1, -5, 10, -10, 5, -1, 1, -5, -4, 25, -4, -5, 1, 1, 15, 64, 50, -50, -64, -15, -1, 1, 15, 65, 66, 30, 66, 65, 15, 1, 1, -55, 455, -671, 1410, -1410, 671, -455, 55, -1, 1, -55, 1815, -4730, 11495, -7251, 11495, -4730, 1815, -55

Offset: 0

Roger L. Bagula, Nov 20 2009

Row sums are:
{1, 0, 1, 0, 4, 0, 9, 0, 324, 0, 9801, 0,...}
Example Matrix F(3):
{{1, 1, 1},
{-1, -3, -2},
{1, 2, 1}}

			{1},
{1, -1},
{1, -1, 1},
{1, 1, -1, -1},
{1, 1, 0, 1, 1},
{1, -5, 10, -10, 5, -1},
{1, -5, -4, 25, -4, -5, 1},
{1, 15, 64, 50, -50, -64, -15, -1},
{1, 15, 65, 66, 30, 66, 65, 15, 1},
{1, -55, 455, -671, 1410, -1410, 671, -455, 55, -1},
{1, -55, 1815, -4730, 11495, -7251, 11495, -4730, 1815, -55, 1},
{1, 197, 4675, -33825, -54978, 99174, -99174, 54978, 33825, -4675, -197, -1}

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 172.

A045912

Clear[T, M, F];
T[n_, m_] := If[n >= m, Binomial[n, m], 0];
M[n_] := Table[T[k, m], {k, 0, n}, {m, 0, n}];
F[n_] := Inverse[Transpose[M[n]]].M[n];
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[F[n], x], x], {n, 0, 10}]];
Flatten[%]

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A123948 Triangle read by rows: first row is 1, and n-th row (n > 0) gives the coefficients in the expansion of the characteristic polynomial of the (n - 1)-th Bernstein basis matrix, horizontal flipped.

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A168228 Coefficient triangle sequence of characteristic polynomials of a Fermat like matrix:M(n)=Pascal n-th matrix: F(n)=Inverse[Transpose[M(n)]].M(n).

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