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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A210236 Triangle of coefficients of polynomials v(n,x) jointly generated with A210235; see the Formula section.

Original entry on oeis.org

1, 3, 2, 6, 8, 3, 11, 22, 16, 4, 19, 52, 57, 28, 5, 32, 112, 166, 124, 45, 6, 53, 228, 428, 432, 241, 68, 7, 87, 446, 1018, 1300, 984, 432, 98, 8, 142, 848, 2285, 3540, 3397, 2036, 728, 136, 9, 231, 1578, 4912, 8964, 10443, 7962, 3914, 1168, 183, 10
Offset: 1

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

Row sums: powers of 3
Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
3....2
6....8....3
11...22...16...4
19...52...57...28...5
First three polynomials v(n,x): 1, 3 + 2x , 6 + 8x + 3x^2.

Crossrefs

Cf. A210235, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]      (* A210235 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]      (* A210236 *)

Formula

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

A210557 Triangle of coefficients of polynomials u(n,x) jointly generated with A210558; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 10, 12, 1, 5, 16, 30, 29, 1, 6, 23, 56, 87, 70, 1, 7, 31, 91, 185, 245, 169, 1, 8, 40, 136, 334, 584, 676, 408, 1, 9, 50, 192, 546, 1158, 1784, 1836, 985, 1, 10, 61, 260, 834, 2052, 3850, 5312, 4925, 2378, 1, 11, 73, 341, 1212, 3366
Offset: 1

Views

Author

Clark Kimberling, Mar 22 2012

Keywords

Comments

Row sums: powers of 3 (see A000244).
For a discussion and guide to related arrays, see A208510.
Subtriangle of (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 23 2012
Up to reflection at the vertical axis, this triangle coincides with the triangle given in A164981, i.e., the numbers are the same just read row-wise in the opposite direction. - Christine Bessenrodt, Jul 20 2012

Examples

			First five rows:
  1;
  1, 2;
  1, 3,  5;
  1, 4, 10, 12;
  1, 5, 16, 30, 29;
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 5x^2.
From _Philippe Deléham_, Mar 23 2012: (Start)
(1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, ...) begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  5,  0;
  1, 4, 10, 12,  0;
  1, 5, 16, 30, 29, 0; (End)

Crossrefs

Cf. A210558, A208510, A164981.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
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[%]   (* A210557 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210558 *)

Formula

u(n,x) = x*u(n-1,x) + x*v(n-1,x)+1,
v(n,x) = 2x*u(n-1,x) + (x+1)v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 23 2012. (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 - 2*y*x + y*x^2 - y^2*x^2)/(1 - x - 2*y*x + y*x^2 - y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)

A210559 Triangle of coefficients of polynomials u(n,x) jointly generated with A210560; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 8, 5, 1, 8, 15, 17, 8, 1, 10, 24, 38, 33, 13, 1, 12, 35, 70, 86, 63, 21, 1, 14, 48, 115, 180, 187, 117, 34, 1, 16, 63, 175, 330, 437, 390, 214, 55, 1, 18, 80, 252, 553, 882, 1007, 791, 386, 89, 1, 20, 99, 348, 868, 1610, 2219, 2235, 1567
Offset: 1

Views

Author

Clark Kimberling, Mar 22 2012

Keywords

Comments

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

Examples

			First five rows:
1
1...2
1...4...3
1...6...8....5
1...8...15...17...8
First three polynomials u(n,x): 1, 1 + 2x, 1 + 4x + 3x^2.

Crossrefs

Cf. A210560, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
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[%]  (* A210559 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210560 *)

Formula

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

A210560 Triangle of coefficients of polynomials v(n,x) jointly generated with A210559; see the Formula section.

Original entry on oeis.org

1, 3, 1, 5, 4, 2, 7, 9, 9, 3, 9, 16, 23, 16, 5, 11, 25, 46, 48, 30, 8, 13, 36, 80, 110, 101, 54, 13, 15, 49, 127, 215, 257, 203, 97, 21, 17, 64, 189, 378, 552, 570, 401, 172, 34, 19, 81, 268, 616, 1057, 1337, 1228, 776, 303, 55, 21, 100, 366, 948, 1862, 2772
Offset: 1

Views

Author

Clark Kimberling, Mar 22 2012

Keywords

Comments

Column 1: odd positive integers (A005408)
Column 2: squares (A000290)
Row n ends with F(n), where F=A000045 (Fibonacci numbers)
Row sums: A005409
Alternating row sums: 1,2,3,4,5,6,7,8,...(A000027)
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
3...1
5...4...2
7...9...9...3
9...16...23...16...5
First three polynomials v(n,x): 1, 3 + x , 5 + 4x + 2x^2.

Crossrefs

Cf. A210559, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
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[%]  (* A210559 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210560 *)

Formula

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

A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

Original entry on oeis.org

1, 3, 2, 6, 9, 4, 10, 25, 24, 8, 15, 55, 85, 60, 16, 21, 105, 231, 258, 144, 32, 28, 182, 532, 833, 728, 336, 64, 36, 294, 1092, 2241, 2720, 1952, 768, 128, 45, 450, 2058, 5301, 8361, 8280, 5040, 1728, 256, 55, 660, 3630, 11385, 22363, 28610, 23920
Offset: 1

Views

Author

Clark Kimberling, Mar 25 2012

Keywords

Comments

Column 1: triangular numbers, A000217
Coefficient of v(n,x): 2^(n-1)
Row sums: A035344
Alternating row sums: 1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Appears to be the reversed row polynomials of A165241 with the unit diagonal removed. If so, the o.g.f. is [1-(1+y)x]/[1-2(1+y)x+(1+y)x^2] - 1/(1-x) and the triangular matrix here may be formed by adding each column of the matrix of A056242, presented in the example section with the additional zeros, to its subsequent column with the first row ignored. - Tom Copeland, Jan 09 2017

Examples

			First five rows:
1
3....2
6....9....4
10...25...24...8
15...55...85...60...16
First three polynomials v(n,x): 1, 3 + 2x, 6 + 9x +4x^2

Crossrefs

Cf. A210753, A208510.
Cf. A056242, A165241.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]    (* A210753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210754 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)

Formula

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

A210806 Triangle of coefficients of polynomials v(n,x) jointly generated with A210805; see the Formula section.

Original entry on oeis.org

1, 0, 2, 1, 1, 3, 0, 3, 3, 5, 1, 2, 8, 7, 8, 0, 4, 8, 19, 15, 13, 1, 3, 15, 25, 42, 30, 21, 0, 5, 15, 46, 67, 89, 58, 34, 1, 4, 24, 58, 128, 164, 182, 109, 55, 0, 6, 24, 90, 186, 330, 378, 363, 201, 89, 1, 5, 35, 110, 300, 536, 804, 833, 709, 365, 144, 0, 7, 35, 155
Offset: 1

Views

Author

Clark Kimberling, Mar 27 2012

Keywords

Comments

Row n ends with F(n), where F=A000045 (Fibonacci numbers).
Column 1: 1,0,1,0,1,0,1,0,...
Alternating row sums: signed Fibonacci numbers.
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
0...2
1...1...3
0...3...3...5
1...2...8...7...8
First three polynomials v(n,x): 1, 2x, 1 + x + 3x^2

Crossrefs

Cf. A210805, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = -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[%]   (* A210805 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210806 *)

Formula

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

A210866 Triangle of coefficients of polynomials u(n,x) jointly generated with A210867; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 2, 1, 10, 21, 14, 3, 1, 15, 55, 65, 31, 5, 1, 21, 120, 235, 187, 65, 8, 1, 28, 231, 700, 867, 503, 134, 13, 1, 36, 406, 1792, 3332, 2914, 1279, 268, 21, 1, 45, 666, 4074, 10955, 13882, 9084, 3122, 527, 34, 1, 55, 1035, 8430, 31563
Offset: 1

Views

Author

Clark Kimberling, Mar 29 2012

Keywords

Comments

Row n starts with 1, followed by the n-th triangular number, and ends with the n-th Fibonacci number.
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...1
1...3....1
1...6....6....2
1...10...21...14...3
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + x^2.

Crossrefs

Cf. A210867, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - x;
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[%]   (* A210866 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210867 *)

Formula

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

A210867 Triangle of coefficients of polynomials v(n,x) jointly generated with A210866; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 5, 2, 4, 15, 12, 3, 5, 34, 51, 28, 5, 6, 65, 170, 156, 60, 8, 7, 111, 465, 680, 438, 126, 13, 8, 175, 1092, 2465, 2411, 1145, 255, 21, 9, 260, 2282, 7623, 10968, 7805, 2854, 506, 34, 10, 369, 4356, 20608, 42735, 43440, 23509, 6813, 984, 55
Offset: 1

Views

Author

Clark Kimberling, Mar 29 2012

Keywords

Comments

For n>1, row n starts with n and ends with F(n), where F=A000045 (Fibonacci numbers).
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...1
3...5....2
4...15...12...3
5...34...51...28...5
First three polynomials v(n,x): 1, 2 + x, 3 + 5x + 2x^2

Crossrefs

Cf. A210866, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - x;
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[%]   (* A210866 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210867 *)

Formula

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

A210873 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 1, 2, 8, 5, 1, 1, 2, 6, 17, 6, 1, 1, 2, 5, 18, 31, 7, 1, 1, 2, 5, 14, 47, 51, 8, 1, 1, 2, 5, 13, 41, 107, 78, 9, 1, 1, 2, 5, 13, 35, 115, 218, 113, 10, 1, 1, 2, 5, 13, 34, 98, 296, 407, 157, 11, 1, 1, 2, 5, 13, 34, 90, 276, 695, 709, 211, 12
Offset: 1

Views

Author

Clark Kimberling, Mar 29 2012

Keywords

Comments

Column 1: 1,1,1,1,1,1,1,1,1...
Row sums: A083318 (1+2^n)
Alternating row sums: A137470
Limiting row: 1,1,2,5,13,34,..., odd-indexed Fibonacci numbers
If the term in row n and column k is written as U(n,k), then U(n,n-1)=A105163.
For a discussion and guide to related arrays, see A208510.

Examples

			First six rows:
1
1...2
1...1...3
1...1...3....4
1...1...2....8...5
1...1...2....6...17...6
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2

Crossrefs

Cf. A210872, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
v[n_, x_] := 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[%]    (* A210872 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210873 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)

Formula

For a discussion and guide to related arrays, see A208510.
u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210874 Triangular array U(n,k) of coefficients of polynomials defined in Comments.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 4, 7, 7, 7, 5, 9, 10, 12, 11, 6, 11, 13, 17, 19, 18, 7, 13, 16, 22, 27, 31, 29, 8, 15, 19, 27, 35, 44, 50, 47, 9, 17, 22, 32, 43, 57, 71, 81, 76, 10, 19, 25, 37, 51, 70, 92, 115, 131, 123, 11, 21, 28, 42, 59, 83, 113, 149, 186, 212, 199, 12, 23, 31
Offset: 1

Views

Author

Clark Kimberling, Mar 30 2012

Keywords

Comments

Polynomials u(n,k) are defined by u(n,x)=x*u(n-1,x)+(x^2)*u(n-2,x)+n*(x+1), where u(1)=1 and u(2,x)=3x+2. The array (U(n,k)) is defined by rows:
u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).
In each column, the first number is a Lucas number and the difference between each two consecutive terms is a Fibonacci number (see the Formula section).
Alternating row sums: 1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)

Examples

			First six rows:
  1
  2...3
  3...5...4
  4...7...7....7
  5...9...10...12...11
  6...11..13...17...19...18
First three polynomials u(n,x): 1, 2 + 3x, 3 + 5x + 4x^2.

Crossrefs

Cf. A208510, A210881, A210875, A210880.

Programs

  • Mathematica
    u[1, x_] := 1; u[2, x_] := 3 x + 2; z = 14;
u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);
Table[Expand[u[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]     (* A210874 *)

Formula

Column k consists of the partial sums of the following sequence: L(k), F(k+1), F(k+1), F(k+1), F(k+1),..., where L=A000032 (Lucas numbers) and F=A000045 (Fibonacci numbers). That is, U(n+1,k)-U(n,k)=F(k+1) for n>1.
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