A303872 -id:A303872 - cp's OEIS frontend

A305098 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

1, -1, 1, 2, -1, -4, 1, 6, 4, -1, -8, -12, 1, 10, 24, 8, -1, -12, -40, -32, 1, 14, 60, 80, 16, -1, -16, -84, -160, -80, 1, 18, 112, 280, 240, 32, -1, -20, -144, -448, -560, -192, 1, 22, 180, 672, 1120, 672, 64, -1, -24, -220, -960, -2016, -1792, -448

Offset: 0

Shara Lalo, May 25 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2*x)^n).
The coefficients in the expansion of 1/(1+x-2x^2) are given by the sequence generated by the row sums.
When n is even the numbers in the row are positive, and when n is odd the numbers in the row are negative.

			Triangle begins:
   1;
  -1;
   1,   2;
  -1,  -4;
   1,   6,    4;
  -1,  -8,  -12;
   1,  10,   24,     8;
  -1, -12,  -40,   -32;
   1,  14,   60,    80,     16;
  -1, -16,  -84,  -160,    -80;
   1,  18,  112,   280,    240,     32;
  -1, -20, -144,  -448,   -560,   -192;
   1,  22,  180,   672,   1120,    672,     64;
  -1, -24, -220,  -960,  -2016,  -1792,   -448;
   1,  26,  264,  1320,   3360,   4032,   1792,    128;
  -1, -28, -312, -1760,  -5280,  -8064,  -5376,  -1024;
   1,  30,  364,  2288,   7920,  14784,  13440,   4608,   256;
  -1, -32, -420, -2912, -11440, -25344, -29568, -15360, -2304;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 389-391.

Signed version of A128099.
Row sums give A077925.
Cf. A303872, A033999 (column 0).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, -t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -T(n-1, k) + 2*T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 26 2018

G.f.: 1 / (1 + t*x - 2t^2).

A317506 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 16, -1, 32, -4, 64, -12, 128, -32, 256, -80, 1, 512, -192, 6, 1024, -448, 24, 2048, -1024, 80, 4096, -2304, 240, -1, 8192, -5120, 672, -8, 16384, -11264, 1792, -40, 32768, -24576, 4608, -160, 65536, -53248, 11520, -560, 1, 131072, -114688, 28160, -1792, 10

Offset: 0

Shara Lalo, Aug 31 2018

The numbers in rows of the triangle are along "third layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "third layer" skew diagonals pointing top-left in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1-2x+x^4) are given by the sequence generated by the row sums. The row sums give A008937. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.83928675521416113... (A058265: Decimal expansion of the tribonacci constant t, the real root of x^3-x^2-x-1), when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8;
      16,      -1;
      32,      -4;
      64,     -12;
     128,     -32;
     256,     -80,     1;
     512,    -192,     6;
    1024,    -448,    24;
    2048,   -1024,    80;
    4096,   -2304,   240,    -1;
    8192,   -5120,   672,    -8;
   16384,  -11264,  1792,   -40;
   32768,  -24576,  4608,  -160;
   65536,  -53248, 11520,  -560,  1;
  131072, -114688, 28160, -1792, 10;
  262144, -245760, 67584, -5376, 60;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Row sums give A008937.
Cf. A065109, A303872.
Cf. A133156, A305098.
Cf. A058265.

t[n_, k_] := t[n, k] = 2^(n - 4 k) * (-1)^k/((n - 4 k)! k!) * (n - 3 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]}] // Flatten

T(n,k) = 2^(n - 4*k) * (-1)^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).

A317504 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, -1, 16, -4, 32, -12, 64, -32, 1, 128, -80, 6, 256, -192, 24, 512, -448, 80, -1, 1024, -1024, 240, -8, 2048, -2304, 672, -40, 4096, -5120, 1792, -160, 1, 8192, -11264, 4608, -560, 10, 16384, -24576, 11520, -1792, 60, 32768, -53248, 28160, -5376, 280, -1, 65536, -114688, 67584, -15360, 1120, -12

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-left in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1-2x+x^3) are given by the sequence generated by the row sums. The row sums give A000071 (Fibonacci numbers - 1). If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.61803398874989484... (A001622: Decimal expansion of Golden ratio (phi or tau) = (1 + sqrt(5))/2), when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8,      -1;
      16,      -4;
      32,     -12;
      64,     -32,      1;
     128,     -80,      6;
     256,    -192,     24;
     512,    -448,     80,      -1;
    1024,   -1024,    240,      -8;
    2048,   -2304,    672,     -40;
    4096,   -5120,   1792,    -160,     1;
    8192,  -11264,   4608,    -560,    10;
   16384,  -24576,  11520,   -1792,    60;
   32768,  -53248,  28160,   -5376,   280,   -1;
   65536, -114688,  67584,  -15360,  1120,  -12;
  131072, -245760, 159744,  -42240,  4032,  -84;
  262144, -524288, 372736, -112640, 13440, -448, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

Row sums give A000071.
Cf. A065109, A303872.
Cf. A133156, A305098.
Cf. A001622.

t[n_, k_] := t[n, k] = 2^(n - 3k) * (-1)^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = 2^(n - 3k) * (-2)^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -1, 1, -1, 2, 1, -4, -1, 6, 1, -8, 4, -1, 10, -12, 1, -12, 24, -1, 14, -40, 8, 1, -16, 60, -32, -1, 18, -84, 80, 1, -20, 112, -160, 16, -1, 22, -144, 280, -80, 1, -24, 180, -448, 240, -1, 26, -220, 672, -560, 32, 1, -28, 264, -960, 1120, -192, -1, 30, -312, 1320, -2016, 672, 1, -32, 364, -1760, 3360, -1792, 64, -1, 34, -420, 2288, -5280, 4032, -448

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1+x+2x^3) are given by the sequence generated by the row sums (see A077973).

			Triangle begins:
   1;
  -1;
   1;
  -1,   2;
   1,  -4;
  -1,   6;
   1,  -8,    4;
  -1,  10,  -12;
   1, -12,   24;
  -1,  14,  -40,     8;
   1, -16,   60,   -32;
  -1,  18,  -84,    80;
   1, -20,  112,  -160,    16;
  -1,  22, -144,   280,   -80;
   1, -24,  180,  -448,   240;
  -1,  26, -220,   672,  -560,    32;
   1, -28,  264,  -960,  1120,  -192;
  -1,  30, -312,  1320, -2016,   672;
   1, -32,  364, -1760,  3360, -1792,   64;
  -1,  34, -420,  2288, -5280,  4032, -448;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

Row sums give A077973.
Cf. A065109, A303872.
Cf. A133156, A305098.

t[n_, k_] := t[n, k] = (-1)^(n - 3k) * 2^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, - t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = (-1)^(n - 3k) * 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A317509 Coefficients in expansion of 1/(1 + x - 2*x^5).

Original entry on oeis.org

1, -1, 1, -1, 1, 1, -3, 5, -7, 9, -7, 1, 9, -23, 41, -55, 57, -39, -7, 89, -199, 313, -391, 377, -199, -199, 825, -1607, 2361, -2759, 2361, -711, -2503, 7225, -12743, 17465, -18887, 13881, 569, -26055, 60985, -98759, 126521

Offset: 0

Shara Lalo, Sep 04 2018

Coefficients in expansion of 1/(1 + x - 2*x^5) are given by the sum of numbers along "fourth Layer" skew diagonals pointing top-left in triangle A065109 ((2-x)^n) and by the sum of numbers along "fourth Layer" skew diagonals pointing top-right in triangle A303872 ((-1+2*x)^n), see links.

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Shara Lalo, Fourth layer skew diagonals in center-justified triangle of coefficients in expansion of (2 - x)^n
Shara Lalo, Fourth layer skew diagonals in center-justified triangle of coefficients in expansion of (-1 + 2x)^n
Index entries for linear recurrences with constant coefficients, signature (-1,0,0,0,2).

Cf. A065109, A303872.

CoefficientList[Series[1/(1 + x - 2 x^5), {x, 0, 42}], x]
a[0] = 1; a[n_] := a[n] = If[n < 0, 0, - a[n - 1] + 2 * a[n - 5]]; Table[a[n], {n, 0, 42}] // Flatten
LinearRecurrence[{-1,0,0,0,2}, {1,-1,1,-1,1}, 43]

my(x='x+O('x^99)); Vec(1/(1+x-2*x^5)) \\ Altug Alkan, Sep 04 2018

a(0)=1, a(n) = -1 * a(n-1) + 2 * a(n-5) for n >= 0; a(n)=0 for n < 0.

cp's OEIS Frontend

A305098 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

A317506 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A317504 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A317509 Coefficients in expansion of 1/(1 + x - 2*x^5).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula