cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A367297 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.

Original entry on oeis.org

1, 2, 3, 5, 10, 8, 12, 34, 38, 21, 29, 104, 161, 130, 55, 70, 305, 592, 654, 420, 144, 169, 866, 2023, 2788, 2436, 1308, 377, 408, 2404, 6556, 10810, 11756, 8574, 3970, 987, 985, 6560, 20446, 39164, 50779, 46064, 28987, 11822, 2584, 2378, 17663, 61912, 134960, 202630, 218717, 171232, 95078, 34690, 6765
Offset: 1

Views

Author

Clark Kimberling, Nov 26 2023

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
    1
    2    3
    5   10    8
   12   34   38    21
   29  104  161   130    55
   70  305  592   654   420  144
  169  866 2023  2788  2436 1308  377
  408 2404 6556 10810 11756 8574 3970 987
Row 4 represents the polynomial p(4,x) = 12 + 34*x + 38*x^2 + 21*x^3, so (T(4,k)) = (12,34,38,21), k=0..3.

Links

Crossrefs

Cf. A000129 (column 1), A001906 (p(n,n-1)), A107839 (row sums, p(n,1)), A077925 (alternating row sums, p(n,-1)), A023000 (p(n,2)), A001076 (p(n,-2)), A186446 (p(n,-3)), A094440, A367208, A367209, A367210, A367211, A367298, A367299, A367300, A367301.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 2 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 2 + 3*x, u = p(2,x), and v = 1 - 2*x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 4*x + 5*x^2)), b = (1/2)*(3*x + 2 + 1/k), c = (1/2)*(3*x + 2 - 1/k).

A367298 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 4*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.

Original entry on oeis.org

1, 2, 4, 5, 14, 15, 12, 48, 76, 56, 29, 148, 326, 372, 209, 70, 436, 1212, 1904, 1718, 780, 169, 1242, 4169, 8228, 10191, 7642, 2911, 408, 3456, 13576, 32176, 49992, 51488, 33112, 10864, 985, 9448, 42492, 117304, 218254, 281976, 249612, 140712, 40545
Offset: 1

Views

Author

Clark Kimberling, Nov 26 2023

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
    1
    2     4
    5    14     15
   12    48     76     56
   29   148    326    372    209
   70   436   1212   1904   1718   780
  169  1242   4169   8228  10191  7642    2911
  408  3456  13576  32176  49992  51488  33112  10864
Row 4 represents the polynomial p(4,x) = 12 + 48*x + 76*x^2 + 56*x^3, so (T(4,k)) = (12,48,76,56), k=0..3.

Links

Crossrefs

Cf. A000129 (column 1), A001353 (p(n,n-1)), A154244 (row sums, p(n,1)), A002605 (alternating row sums, p(n,-1)), A190989 (p(n,2)), A005668 (p(n,-2)), A190869 (p(n,-3)), A094440, A367208, A367209, A367210, A367211, A367297, A367299, A367300, A367301.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 2 + 4 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 2 + 4*x, u = p(2,x), and v = 1 - 2*x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 8*x + 12*x^2)), b = (1/2)*(4*x + 2 + 1/k), c = (1/2)*(4*x + 2 - 1/k).

A368518 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 3*x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 7, 3, 10, 18, 20, 5, 20, 51, 68, 61, 8, 40, 118, 220, 251, 182, 13, 76, 264, 584, 905, 888, 547, 21, 142, 558, 1452, 2678, 3540, 3076, 1640, 34, 260, 1145, 3380, 7279, 11536, 13418, 10456, 4921, 55, 470, 2286, 7548, 18391, 33990, 47600, 49552
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    2
   2    4    7
   3   10   18    20
   5   20   51    68    61
   8   40  118   220   251   182
  13   76  264   584   905   888   547
  21  142  558  1452  2678  3540  3076  1640
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 18*x^2 + 20*x^3, so (T(4,k)) = (3,10,18,20), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A002605, (p(n,n-1)); A030195 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A015545, (p(n,2)); A099012, (p(n,-2)); A087567, (p(n,3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155, A368156.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 3x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + 32*x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 16*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A368150 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

Original entry on oeis.org

1, 1, 3, 2, 6, 8, 3, 15, 25, 21, 5, 30, 76, 90, 55, 8, 60, 188, 324, 300, 144, 13, 114, 439, 948, 1251, 954, 377, 21, 213, 961, 2529, 4207, 4527, 2939, 987, 34, 390, 2026, 6246, 12606, 17154, 15646, 8850, 2584, 55, 705, 4136, 14640, 34590, 56970, 65840
Offset: 1

Views

Author

Clark Kimberling, Dec 25 2023

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    3
   2    6    8
   3   15   25    21
   5   30   76    90     55
   8   60  188   324    300   144
  13  114  439   948   1251   954   377
  21  213  961  2529   4207  4527  2939   987
Row 4 represents the polynomial p(4,x) = 3 + 15*x + 25*x^2 + 21*x^3, so (T(4,k)) = (3,15,25,21), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A001906 (p(n,n-1)); A000302 (row sums), (p(n,1)); A122803 (alternating row sums), (p(n,-1)); A190972 (p(n,2)), A116415, (p(n,-2)); A190990, (p(n,3)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368151.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 1 - x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).

A368151 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where u = p(2,x), v = 2 - x^2.

Original entry on oeis.org

1, 1, 3, 3, 6, 8, 5, 21, 25, 21, 11, 48, 101, 90, 55, 21, 123, 290, 414, 300, 144, 43, 282, 850, 1416, 1551, 954, 377, 85, 657, 2255, 4671, 6109, 5481, 2939, 987, 171, 1476, 5883, 13986, 22374, 24300, 18585, 8850, 2584, 341, 3303, 14736, 40320, 74295, 97713
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2023

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1     3
   3     6    8
   5    21    25    21
  11    48   101    90    55
  21   123   290   414   300  144
  43   282   850  1416  1551  954    377
  85   657  2255  4671  6109  5481  2939  987
Row 4 represents the polynomial p(4,x) = 5 + 21 x + 25 x^2 + 21 x^3, so (T(4,k)) = (5,21,25,21), k=0..3.

Links

Crossrefs

Cf. A001045 (column 1); A001906 (p(n,n-1)); A001076 (row sums), (p(n,1)); A077985 (alternating row sums), (p(n,-1)); A186446 (p(n,2)), A107839, (p(n,-2)); A190989, (p(n,3)); A023000, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 2 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where p(1,x) = 1, p(2,x) = 1 + 3 x, u = p(2,x), and v = 2 - x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(9 + 6 x + 5 x^2), b = (1/2) (3 x + 1 - 1/k), c = (1/2) (3 x + 1 + 1/k).

A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 7, 27, 25, 21, 19, 66, 126, 90, 55, 40, 204, 392, 504, 300, 144, 97, 522, 1363, 1884, 1851, 954, 377, 217, 1425, 4065, 7281, 8011, 6435, 2939, 987, 508, 3642, 12332, 24606, 34044, 31446, 21524, 8850, 2584, 1159, 9441, 35236, 82020, 127830
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
    1
    1    3
    4    6    8
    7   27   25   21
   19   66  126   90   55
   40  204  392  504  300  144
   97  522 1363 1884 1851  954  377
  217 1425 4065 7281 8011 6435 2939 987
Row 4 represents the polynomial p(4,x) = 7 + 27*x + 25*x^2 + 21*x^3, so (T(4,k)) = (7,27,25,21), k=0..3.

Links

Crossrefs

Cf. A006130 (column 1); A001906 (p(n,n-1)); A090017 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004187, (p(n,2)); A004254, (p(n,-2)); A190988, (p(n,3)); A190978 (unsigned), (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 3 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 3 - x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(13 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).

A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - x^2.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 4, -2, 4, 5, 5, 4, -10, 5, 8, 10, -3, 4, -25, 6, 13, 16, 1, -29, 14, -49, 7, 21, 28, -8, -24, -78, 56, -84, 8, 34, 47, -12, -88, -26, -162, 168, -132, 9, 55, 80, -31, -140, -200, 100, -330, 408, -195, 10, 89, 135, -58, -301, -230, -296
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1   2
   2   1   3
   3   4  -2    4
   5   5   4  -10    5
   8  10  -3    4  -25    6
  13  16   1  -29   14  -49    7
  21  28  -8  -24  -78   56  -84   8
Row 4 represents the polynomial p(4,x) = 3 + 4*x - 2*x^2 + 4*x^3, so (T(4,k)) = (3,4,-2,4), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A000027 (p(n,n-1)); A057083 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A190970, (p(n,2)); A030195, (p(n,-2)); A052918, (p(n,-3)); A190972, (p(n,-4)); A057085, (p(n,-5)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 - 3*x - x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 8*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A368155 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - 2*x^2.

Original entry on oeis.org

1, 1, 3, 2, 3, 7, 3, 9, 5, 15, 5, 15, 26, 3, 31, 8, 30, 43, 63, -15, 63, 13, 54, 104, 87, 144, -81, 127, 21, 99, 203, 273, 115, 333, -275, 255, 34, 177, 416, 549, 609, -9, 806, -789, 511, 55, 315, 811, 1263, 1146, 1260, -725, 2043, -2071, 1023, 89, 555, 1573
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    3
   2    3     7
   3    9     5    15
   5   15    26     3    31
   8   30    43    63   -15    63
  13   54   104    87   144   -81    127
  21   99   203   273   115   333   -275   255
Row 4 represents the polynomial p(4,x) = 3 + 9*x + 5*x^2 + 15*x^3, so (T(4,k)) = (3,9,5,15), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A000225, (p(n,n-1)); A001787 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004254, (p(n,-2)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368156.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 1 - 3*x - 2*x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 6*x + x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A368156 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 3, 10, 14, 12, 5, 20, 41, 44, 29, 8, 40, 98, 148, 131, 70, 13, 76, 224, 408, 497, 376, 169, 21, 142, 482, 1044, 1542, 1588, 1052, 408, 34, 260, 1003, 2492, 4351, 5456, 4894, 2888, 985, 55, 470, 2026, 5684, 11359, 16790, 18400, 14672, 7813
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    2
   2    4    5
   3   10   14    12
   5   20   41    44    29
   8   40   98   148   131    70
  13   76  224   408   497   376   169
  21  142  482  1044  1542  1588  1052  408
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 14*x^2 + 12*x^3, so (T(4,k)) = (3,10,14,12), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A000129, (p(n,n-1)); A007482 (row sums), (p(n,1)); A077925 (alternating row sums), (p(n,-1)); A057088, (p(n,2)); A015523, (p(n,-2)); A015568, (p(n,3)); A180250, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 8*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A368149 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 3, 10, 10, 4, 5, 20, 31, 20, 5, 8, 40, 78, 76, 35, 6, 13, 76, 184, 232, 161, 56, 7, 21, 142, 406, 636, 582, 308, 84, 8, 34, 260, 861, 1604, 1831, 1296, 546, 120, 9, 55, 470, 1766, 3820, 5215, 4630, 2640, 912, 165, 10, 89, 840, 3533, 8696
Offset: 1

Views

Author

Clark Kimberling, Dec 25 2023

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    2
   2    4    3
   3   10   10    4
   5   20   31   20    5
   8   40   78   76   35    6
  13   76  184  232  161   56   7
  21  142  406  636  582  308  84  8
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 10*x^2 + 4*x^3, so (T(4,k)) = (3,10,10,4), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A000027 (p(n,n-1)); A000244 (row sums), (p(n,1)); A033999 (alternating row sums), (p(n,-1)); A116415 (p(n,2)), A000748, (p(n,-2)); A152268, (p(n,3)); A190969, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 - x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).
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