A329433 -id:A329433 - cp's OEIS frontend

A329429 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 1, 2, 2, 1, 5, 8, 8, 4, 1, 26, 80, 144, 168, 138, 80, 32, 8, 1, 677, 4160, 13888, 31776, 54792, 74624, 82432, 74944, 56472, 35296, 18208, 7664, 2580, 672, 128, 16, 1, 458330, 5632640, 36109952, 158572864, 531441232, 1439520512, 3264101376, 6342205824

Offset: 0

Clark Kimberling, Nov 13 2019

Let f(x) = x^2 + 1, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Except for the first term, the sequence (p(n,0)) = (1, 1, 5, 26, 677, ...) is found in A003095 and A008318. This is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..4:
   1;
   1,  1;
   2,  2,   1;
   5,  8,   8,   4,   1;
  26, 80, 144, 168, 138, 80, 32, 8, 1.
Rows 0..4, the polynomials u(n,x):
  1,
  1 + x^2,
  2 + 2 x^2 + x^4,
  5 + 8 x^2 + 8 x^4 + 4 x^6 + x^8,
  26 + 80 x^2 + 144 x^4 + 168 x^6 + 138 x^8 + 80 x^10 + 32 x^12 + 8 x^14 + x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A003095, A008318, A329430, A329431, A329432, A329433.

f[x_] := x^2 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329429 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 7}]  (* A329429 array *)

p(n,0)  = (1, 1, 2, 5, 26, 677, 458330, ...)
p(n,1)  =    (1, 2, 5, 26, 677, 458330, ...)
p(n,2)  =       (2, 5, 26, 677, 458330, ...)
p(n,5)  =          (5, 26, 677, 458330, ...)
p(n,26) =             (26, 677, 458330, ...), etc.;
that is, p(n,p(k,0)) = p(n+k-2,0); there are similar identities for other sequences p(n,h).

A329432 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 19, 96, 224, 256, 128, 723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768, 1045459, 21100032, 209001984, 1339772928, 6194997248, 21845442560, 60641837056, 134967984128, 243130040320, 355391766528, 419950493696, 396881821696

Offset: 0

Clark Kimberling, Nov 23 2019

Let f(x) = 2 x^2 + 1, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 1, 3, 19, 723, 1045459, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..4:
  1;
  1, 2;
  3, 8, 8;
  19, 96, 224, 256, 128;
  723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768.
Rows 0..4, the polynomials u(n,x):
  1,
  1 + 2 x^2
  3 + 8 x^2 + 8 x^4
  19 + 96 x^2 + 224 x^4 + 256 x^6 + 128 x^8
  723 + 7296 x^2 + 35456 x^4 + 105472 x^6 + 208384 x^8 + 278528 x^10 + 245760 x^12 + 131072 x^14 + 32768 x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329430, A329431, A329433.

f[x_] := 2 x^2 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329432 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329432 array *)

A329430 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 1, 9, 36, 90, 147, 171, 144, 87, 36, 9, 1, 730, 8748, 56862, 257337, 895941, 2528172, 5967108, 12025098, 20984508, 32024268, 43036029, 51168267, 53983503, 50609772, 42164064, 31176036, 20403009, 11768247, 5946156, 2610171, 984420, 314262, 83214, 17766, 2934, 351, 27, 1

Offset: 0

Clark Kimberling, Nov 23 2019

Let f(x) = x^3 + 1, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)).

Then the sequence (p(n,0)) = (1,1,2,9,730, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..3:
  1;
  1, 1;
  2, 3, 3, 1;
  9, 36, 90, 147, 171, 144, 87, 36, 9, 1.
Rows 0..3, the polynomials u(n,x):
  1;
  1 + x^3;
  2 + 3 x^3 + 3 x^6 + x^9;
  9 + 36 x^3 + 90 x^6 + 147 x^9 + 171 x^12 + 144 x^15 + 87 x^18 + 36 x^21 + 9 x^24 + x^27.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329431, A329432, A329433.

f[x_] := x^3 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329430 polynomials u(n,x) *)
Table[CoefficientList[u[n, x^(1/3)], x], {n, 0, 5}]  (* A329430 array *)

A329431 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 38, 48, 28, 8, 1, 1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1, 2090918, 10550016, 26125248, 41867904, 48398416, 42666880, 29610272, 16475584, 7419740, 2711424, 800992, 189248, 35064, 4928, 496, 32, 1, 4371938082726, 44118436709376

Offset: 0

Clark Kimberling, Nov 23 2019

Let f(x) = x^2 + 2, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)).

Then the sequence (p(n,0)) = (1,2,6,38,1446, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889. p(n,0) = A072191(n) for n >= 1.

			Rows 0..4:
  1;
  2, 1;
  6, 4, 1;
  38, 48, 28, 8, 1;
  1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1.
Rows 0..4, the polynomials u(n,x):
  1;
  2 + x^2;
  6 + 4 x^2 + x^4;
  38 + 48 x^2 + 28 x^4 + 8 x^6 + x^8;
  1446 + 3648 x^2 + 4432 x^4 + 3296 x^6 + 1628 x^8 + 544 x^10 + 120 x^12 + 16 x^14 + x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A329429, A329430, A329432, A329433.

f[x_] := x^2 + 2;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329431 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329431 array *)

A329471 a(n) = a(n-1)^2 + 3 for n >=2 , where a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 12, 147, 21612, 467078547, 218162369067631212, 47594819277201331861096436836588947, 2265266822029455509816214491130485582138030749246532017266850242568812

Offset: 0

Clark Kimberling, Dec 31 2019

nonn

(a(n)) is a strong divisibility sequence.

Cf. A329433.

f[x_] := x^2 + 3; u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Table [u[n, x] /. x -> 0, {n, 0, 10}]

a(n) = p(n,0), where p(n,x) is polynomial defined in A329433.

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A329429 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329432 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329430 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329431 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

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A329471 a(n) = a(n-1)^2 + 3 for n >=2 , where a(0) = 1, a(1) = 3.

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