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.

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A208911 Triangle of coefficients of polynomials u(n,x) jointly generated with A208912; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 1, 12, 14, 8, 1, 20, 32, 38, 16, 1, 30, 60, 110, 90, 32, 1, 42, 100, 250, 300, 214, 64, 1, 56, 154, 490, 770, 826, 490, 128, 1, 72, 224, 868, 1680, 2408, 2128, 1110, 256, 1, 90, 312, 1428, 3276, 5880, 6888, 5382, 2474, 512, 1, 110, 420
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...2
1...6....4
1...12...14...8
1...20...32...38...16
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 4x^2
1 + 12x + 14x^2 + 8x^3
1 + 20x + 32x^2 + 38x^3 + 16x^4

Crossrefs

Cf. A208912, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208911 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208912 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208912 Triangle of coefficients of polynomials v(n,x) jointly generated with A208911; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 5, 4, 4, 9, 15, 8, 5, 14, 36, 37, 16, 6, 20, 70, 105, 91, 32, 7, 27, 120, 235, 306, 213, 64, 8, 35, 189, 455, 791, 819, 491, 128, 9, 44, 280, 798, 1736, 2380, 2136, 1109, 256, 10, 54, 396, 1302, 3402, 5796, 6924, 5373, 2475, 512, 11, 65, 540
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...2
3...5....3
4...9....15...8
5...14...36...37...16
First five polynomials v(n,x):
1
2 + 2x
3 + 5x + 3x^2
4 + 9x + 15x^2 + 8x^3
5 + 14x + 36x^2 + 37x^3 + 16x^4

Crossrefs

Cf. A208911, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208911 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208912 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208913 Triangle of coefficients of polynomials u(n,x) jointly generated with A208914; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 1, 12, 12, 8, 1, 20, 24, 40, 16, 1, 30, 40, 120, 80, 32, 1, 42, 60, 280, 240, 224, 64, 1, 56, 84, 560, 560, 896, 448, 128, 1, 72, 112, 1008, 1120, 2688, 1792, 1152, 256, 1, 90, 144, 1680, 2016, 6720, 5376, 5760, 2304, 512, 1, 110, 180
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...2
1...6....4
1...12...12...8
1...20...24...40...16
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 4x^2
1 + 12x + 12x^2 + 8x^3
1 + 20x + 24x^2 + 40x^3 + 16x^4

Crossrefs

Cf. A208914, A208510.

Programs

  • Mathematica
    u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208913 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208914 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208915 Triangle of coefficients of polynomials u(n,x) jointly generated with A208916; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 12, 14, 1, 8, 18, 36, 38, 1, 10, 24, 66, 108, 94, 1, 12, 30, 104, 210, 308, 246, 1, 14, 36, 150, 344, 674, 892, 622, 1, 16, 42, 204, 510, 1224, 2098, 2500, 1606, 1, 18, 48, 266, 708, 1990, 4024, 6402, 7052, 4094, 1, 20, 54, 336, 938
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...2
1...4...6
1...6...12...14
1...8...18...36...38
First five polynomials u(n,x):
1
1 + 2x
1 + 4x + 6x^2
1 + 6x + 12x^2 + 14x^3
1 + 8x + 18x^2 + 36x^3 + 38x^4

Crossrefs

Cf. A208914, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208915 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208916 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208916 Triangle of coefficients of polynomials v(n,x) jointly generated with A208915; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 11, 19, 1, 3, 15, 35, 47, 1, 3, 19, 51, 107, 123, 1, 3, 23, 67, 183, 323, 311, 1, 3, 27, 83, 275, 603, 939, 803, 1, 3, 31, 99, 383, 963, 1951, 2723, 2047, 1, 3, 35, 115, 507, 1403, 3411, 6147, 7723, 5259, 1, 3, 39, 131, 647, 1923, 5383
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...3
1...3...7
1...3...11...19
1...3...15...35...47
First five polynomials v(n,x):
1
1 + 3x
1 + 3x + 7x^2
1 + 3x + 11x^2 + 19x^3
1 + 3x + 15x^2 + 35x^3 + 47x^4

Crossrefs

Cf. A208915, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208915 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208916 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208917 Triangle of coefficients of polynomials u(n,x) jointly generated with A208918; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 4, 8, 1, 6, 16, 24, 1, 8, 24, 56, 80, 1, 10, 32, 96, 208, 256, 1, 12, 40, 144, 384, 736, 832, 1, 14, 48, 200, 608, 1472, 2624, 2688, 1, 16, 56, 264, 880, 2496, 5632, 9216, 8704, 1, 18, 64, 336, 1200, 3840, 10112, 21120, 32256, 28160, 1, 20, 72
Offset: 1

Views

Author

Clark Kimberling, Mar 04 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...2
1...4...8
1...6...16...24
1...8...24...56...80
First five polynomials u(n,x):
1
1 + 2x
1 + 4x + 8x^2
1 + 6x + 16x^2 + 24x^3
1 + 8x + 24x^2 + 56x^3 + 80x^4

Crossrefs

Cf. A208918, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208917 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208918 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208918 Triangle of coefficients of polynomials v(n,x) jointly generated with A208917; see the Formula section.

Original entry on oeis.org

1, 1, 4, 1, 4, 12, 1, 4, 16, 40, 1, 4, 20, 64, 128, 1, 4, 24, 88, 240, 416, 1, 4, 28, 112, 368, 896, 1344, 1, 4, 32, 136, 512, 1504, 3264, 4352, 1, 4, 36, 160, 672, 2240, 5952, 11776, 14080, 1, 4, 40, 184, 848, 3104, 9472, 23168, 41984, 45568, 1, 4, 44
Offset: 1

Views

Author

Clark Kimberling, Mar 04 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...4
1...4...12
1...4...16...40
1...4...20...64...128
First five polynomials v(n,x):
1
1 + 4x
1 + 4x + 12x^2
1 + 4x + 16x^2 + 40x^3
1 + 4x + 20x^2 + 64x^3 + 128x^4

Crossrefs

Cf. A208917, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208917 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208918 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208919 Triangle of coefficients of polynomials u(n,x) jointly generated with A208920; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 12, 20, 14, 1, 20, 44, 66, 38, 1, 30, 80, 190, 208, 94, 1, 42, 130, 430, 678, 622, 246, 1, 56, 196, 840, 1708, 2380, 1852, 622, 1, 72, 280, 1484, 3668, 6888, 7928, 5338, 1606, 1, 90, 384, 2436, 7056, 16716, 25344, 25650, 15336
Offset: 1

Views

Author

Clark Kimberling, Mar 04 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...2
1...6....6
1...12...20...14
1...20...44...66...38
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 6x^2
1 + 12x + 20x^2 + 14x^3
1 + 20x + 44x^2 + 66x^3 + 38x^4

Crossrefs

Cf. A208920, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208919 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208920 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208920 Triangle of coefficients of polynomials v(n,x) jointly generated with A208919; see the Formula section.

Original entry on oeis.org

1, 2, 3, 3, 7, 7, 4, 12, 26, 19, 5, 18, 62, 85, 47, 6, 25, 120, 235, 264, 123, 7, 33, 205, 515, 879, 803, 311, 8, 42, 322, 980, 2254, 3038, 2358, 803, 9, 52, 476, 1694, 4914, 8708, 10156, 6865, 2047, 10, 63, 672, 2730, 9576, 20958, 32640, 32877, 19588
Offset: 1

Views

Author

Clark Kimberling, Mar 04 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...3
3...7....7
4...12...26...19
5...18...62...85...47
First five polynomials v(n,x):
1
2 + 3x
3 + 7x + 7x^2
4 + 12x + 26x^2 + 19x^3
5 + 18x + 62x^2 + 85x^3 + 47x^4

Crossrefs

Cf. A208919, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208919 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208920 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208921 Triangle of coefficients of polynomials u(n,x) jointly generated with A208922; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 8, 2, 1, 18, 10, 4, 1, 32, 36, 28, 4, 1, 50, 100, 108, 36, 8, 1, 72, 230, 324, 196, 80, 8, 1, 98, 462, 840, 772, 440, 104, 16, 1, 128, 840, 1960, 2456, 1840, 752, 208, 16, 1, 162, 1416, 4200, 6744, 6464, 3824, 1488, 272, 32, 1, 200, 2250, 8376
Offset: 1

Views

Author

Clark Kimberling, Mar 04 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...2
1...8....2
1...18...10...4
1...32...36...28...4
First five polynomials u(n,x):
1
1 + 2x
1 + 8x + 2x^2
1 + 18x + 10x^2 + 4x^3
1 + 32x + 36x^2 + 28x^3 + 4x^4

Crossrefs

Cf. A208922, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208921 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208922 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Previous Showing 81-90 of 345 results. Next

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