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 16 results. Next

A367208 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 - x^2.

Original entry on oeis.org

1, 1, 3, 2, 5, 8, 3, 13, 19, 21, 5, 25, 59, 65, 55, 8, 50, 137, 231, 210, 144, 13, 94, 316, 623, 834, 654, 377, 21, 175, 677, 1615, 2545, 2859, 1985, 987, 34, 319, 1411, 3859, 7285, 9691, 9451, 5911, 2584, 55, 575, 2849, 8855, 19115, 30245, 35105, 30407, 17345, 6765
Offset: 1

Views

Author

Clark Kimberling, Nov 13 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 ten rows:
   1
   1    3
   2    5     8
   3   13    19    21
   5   25    59    65     55
   8   50   137   231    210    144
  13   94   316   623    834    654    377
  21  175   677  1615   2545   2859   1985    987
  34  319  1411  3859   7285   9691   9451   5911   2584
  55  575  2849  8855  19115  30245  35105  30407  17345  6765
Row 4 represents the polynomial p(4,x) = 3 + 13*x + 19*x^2 + 21*x^3, so (T(4,k)) = (3,13,19,21), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1), A001906 (T(n,n-1)), A001353 (row sums, p(n,1)), A077985 (alternating row sums, p(n,-1)), A190974 (p(n,2)), A004254 (p(n,-2)), A190977 (p(n,-3)), A094440, A367209, A367210, A367211, A367297, A367298, A367299, A367300.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 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) = 1 + 3*x, u = p(2,x), and v = 1 - x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/D), b = (1/2)*(1 + 3*x - D), c = (1/2)*(1 + 3*x + D), where D = sqrt(5 + 2*x + 5*x^2).

A367209 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 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 - x - x^2.

Original entry on oeis.org

1, 1, 4, 2, 7, 15, 3, 18, 38, 56, 5, 35, 116, 186, 209, 8, 70, 273, 650, 859, 780, 13, 132, 629, 1777, 3366, 3821, 2911, 21, 246, 1352, 4600, 10410, 16556, 16556, 10864, 34, 449, 2820, 11024, 29770, 56874, 78504, 70356, 40545, 55, 810, 5701, 25306, 78324
Offset: 1

Views

Author

Clark Kimberling, Nov 13 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 nine rows:
   1
   1    4
   2    7    15
   3   18    38     56
   5   35   116    186    209
   8   70   273    650    859    780
  13  132   629   1777   3366   3821   2911
  21  246  1352   4600  10410  16556  16556  10864
  34  449  2820  11024  29770  56874  78504  70356  405459
Row 4 represents the polynomial p(4,x) = 3 + 18*x + 38*x^2 + 56*x^3, so (T(4,k)) = (3,18,38,56), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1), A001353 (T(n,n-1)), A004254 (row sums, p(n,1)), A006190 (alternating row sums, p(n,-1)), A094440, A367208, A367210, A367211, A367297, A367298, A367299, A367300.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 4 x; u[x_] := p[2, x]; v[x_] := 1 - 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) = 1 + 4*x, u = p(2,x), and v = 1 - x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/D), b = (1/2)*(1 + 4*x - D), c = (1/2)*(1 + 4*x + D), where D = sqrt(5 + 4*x + 12*x^2).

A367210 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 5x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where u = p(2,x), v = 1 - x - x^2.

Original entry on oeis.org

1, 1, 5, 2, 9, 24, 3, 23, 63, 115, 5, 45, 191, 397, 551, 8, 90, 453, 1381, 2358, 2640, 13, 170, 1044, 3807, 9226, 13482, 12649, 21, 317, 2249, 9865, 28785, 58513, 75061, 60605, 34, 579, 4695, 23703, 82485, 202887, 357567, 409779, 290376, 55, 1045, 9501
Offset: 1

Views

Author

Clark Kimberling, Nov 13 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    5
 2    9    24
 3   23    63   115
 5   45   191   397    551
 8   90   453  1381   2358   2640
13  170  1044  3807   9226  13482  12649
21  317  2249  9865  28785  58513  75061  60605
Row 4 represents the polynomial p(4,x) = 3 + 23 x + 63 x^2 + 115 x^3, so  that (T(4,k)) = (3,23,63,115), k-0..3.

Links

Crossrefs

Cf. A000045 (column 1), A004254 (T(n,n-1)), A001109 (row sums p(n,1)), A001076 (alternating row sums, p(n,-1)), A094440, A367208, A367209, A367211, A367297, A367298, A367299, A367300.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 5 x; u[x_] := p[2, x]; v[x_] := 1 - 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) = 1 + 5x, u = p(2,x), and v = 1 - x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/D), b = 1/2 (1 + 5 x - D), c = 1/2 (1 + 5 x + D), where D = sqrt(5 + 6 x + 21 x^2).

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).

A367299 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 5*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, 5, 5, 18, 24, 12, 62, 126, 115, 29, 192, 545, 794, 551, 70, 567, 2040, 4114, 4716, 2640, 169, 1618, 7047, 17940, 28420, 26964, 12649, 408, 4508, 23020, 70582, 140988, 185122, 150122, 60605, 985, 12336, 72222, 258492, 620379, 1027368, 1156155, 819558, 290376
Offset: 1

Views

Author

Clark Kimberling, Dec 23 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    5
    5   18    24
   12   62   126   115
   29  192   545   794    551
   70  567  2040  4114   4716   2640
  169 1618  7047 17940  28420  26964  12649
  408 4508 23020 70582 140988 185122 150122 60605
Row 4 represents the polynomial p(4,x) = 12 + 62*x + 126*x^2 + 115*x^3, so (T(4,k)) = (12,62,126,115), k=0..3.

Links

Crossrefs

Cf. A000129 (column 1); A004254 (p(n,n-1)); A186446 (row sums, p(n,1)); A007482 (alternating row sums, p(n,-1)); A041025 (p(n,-2)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367300.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 2 + 5 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 + 5*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 + 12*x + 21*x^2)), b = (1/2) (5*x + 2 + 1/k), c = (1/2) (5*x + 2 - 1/k).

A367301 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 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, 3, 3, 10, 16, 8, 33, 75, 63, 21, 109, 320, 380, 220, 55, 360, 1296, 1980, 1620, 720, 144, 1189, 5070, 9459, 9940, 6255, 2262, 377, 3927, 19353, 42615, 54561, 44085, 22635, 6909, 987, 12970, 72532, 184034, 277480, 272854, 179972, 78230, 20672, 2584
Offset: 1

Views

Author

Clark Kimberling, Dec 23 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
     3      3
    10     16      8
    33     75     63     21
   109    320    380    220     55
   360   1296   1980   1620    720    144
  1189   5070   9459   9940   6255   2262   377
  3927  19353  42615  54561  44085  22635  6909  987
Row 4 represents the polynomial p(4,x) = 33 + 75*x + 63*x^2 + 21*x^3, so (T(4,k)) = (33,75,63,21), k=0..3.

Links

Crossrefs

Cf. A006190 (column 1); A001906 (p(n,n-1)); A154244 (row sums, p(n,1)); A077957 (alternating row sums, p(n,-1)); A190984 (p(n,2)); A006190 (signed, p(n,-2)); A154244 (p(n,-3)); A190984 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 3 + 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) = 3 + 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(13 + 10*x + 5*x^2)), b = (1/2) (3*x + 3 + 1/k), c = (1/2) (3*x + 3 - 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).
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