A367211 -id:A367211 - cp's OEIS frontend

A368150 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) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

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

Clark Kimberling, Dec 25 2023

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.

			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.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

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.

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}]]

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) = up(n-1,x) + vp(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

Clark Kimberling, Dec 31 2023

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.

			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.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

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.

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}]]

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 + 3x, p(n,x) = up(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

Clark Kimberling, Jan 20 2024

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.

			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.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), 1-28, Paper No. A14.

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.

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}]]

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 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - 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

Clark Kimberling, Jan 20 2024

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.

			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.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), 1-28, Paper No. A14.

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.

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}]]

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 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - 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

Clark Kimberling, Jan 20 2024

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.

			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.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

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.

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}]]

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 + 2x, p(n,x) = up(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

Clark Kimberling, Jan 20 2024

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.

			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.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

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.

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}]]

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).

A367564 Triangular array read by rows: T(n, k) = binomial(n, k) * A001333(n - k).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 17, 28, 18, 4, 1, 41, 85, 70, 30, 5, 1, 99, 246, 255, 140, 45, 6, 1, 239, 693, 861, 595, 245, 63, 7, 1, 577, 1912, 2772, 2296, 1190, 392, 84, 8, 1, 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1, 3363, 13930, 25965, 28680, 20790, 10332, 3570, 840, 135, 10, 1

Offset: 0

Peter Luschny, Nov 25 2023

			Triangle T(n, k) starts:
[0]    1;
[1]    1,    1;
[2]    3,    2,    1;
[3]    7,    9,    3,    1;
[4]   17,   28,   18,    4,    1;
[5]   41,   85,   70,   30,    5,    1;
[6]   99,  246,  255,  140,   45,    6,   1;
[7]  239,  693,  861,  595,  245,   63,   7,   1;
[8]  577, 1912, 2772, 2296, 1190,  392,  84,   8, 1;
[9] 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1;

Cf. A001333 (column 0), A006012 (row sums), A367211.

P := proc(n) option remember; ifelse(n <= 1, 1, 2*P(n - 1) + P(n - 2)) end:
T := (n, k) -> P(n - k) * binomial(n, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

P[n_] := P[n] = If[n <= 1, 1, 2 P[n - 1] + P[n - 2]];
T[n_, k_] := P[n - k] Binomial[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 10 2024, after Peter Luschny *)

From Werner Schulte, Nov 26 2023: (Start)
The row polynomials p(n, x) = Sum_{k=0..n} T(n, k) * x^k satisfy:
  a) p'(n, x) = n * p(n-1, x) where p' is the first derivative of p;
  b) p(0, x) = 1, p(1, x) = 1 + x and p(n, x) = (2+2*x) * p(n-1, x) + (1-2*x-x^2) * p(n-2, x) for n > 1.
T(n, 0) = A001333(n) for n >= 0 and T(n, k) = T(n-1, k-1) * n / k for 0 < k <= n.
G.f.: (1 - (1+x) * t) / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)

A368149 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(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

Clark Kimberling, Dec 25 2023

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.

			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.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

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.

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}]]

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).

A368157 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 2x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 3, 10, 16, 16, 5, 20, 46, 56, 44, 8, 40, 108, 184, 188, 120, 13, 76, 244, 496, 692, 608, 328, 21, 142, 520, 1248, 2088, 2480, 1920, 896, 34, 260, 1074, 2936, 5764, 8256, 8592, 5952, 2448, 55, 470, 2156, 6616, 14764, 24760, 31200, 28992

Offset: 1

Clark Kimberling, Jan 20 2024

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.

			First eight rows:
   1
   1    2
   2    4    6
   3   10   16    16
   5   20   46    56    44
   8   40  108   184   188   120
  13   76  244   496   692   608   328
  21  142  520  1248  2088  2480  1920  896
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 16*x^2 + 16*x^3, so (T(4,k)) = (3,10,16,16), k=0..3.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

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.

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}]]

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 + 2*x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 13*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

cp's OEIS Frontend

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.

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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.

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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.

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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.

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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.

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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.

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A367564 Triangular array read by rows: T(n, k) = binomial(n, k) * A001333(n - k).

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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.

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A368150 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) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

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) = up(n-1,x) + vp(n-2,x) for n >=3, where u = p(2,x), v = 2 - x^2.

A368152 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) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

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

A368155 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) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - 2*x^2.

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

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

A368157 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 2x^2.