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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A208755 Triangle of coefficients of polynomials u(n,x) jointly generated with A208756; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 6, 8, 1, 2, 8, 14, 16, 1, 2, 10, 20, 34, 32, 1, 2, 12, 26, 56, 78, 64, 1, 2, 14, 32, 82, 140, 178, 128, 1, 2, 16, 38, 112, 218, 352, 398, 256, 1, 2, 18, 44, 146, 312, 594, 852, 882, 512, 1, 2, 20, 50, 184, 422, 912, 1530, 2040, 1934, 1024
Offset: 1

Views

Author

Clark Kimberling, Mar 01 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012

Examples

			First five rows:
  1;
  1, 2;
  1, 2, 4;
  1, 2, 6,  8;
  1, 2, 8, 14, 16;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 2x + 4x^2
  1 + 2x + 6x^2 +  8x^3
  1 + 2x + 8x^2 + 14x^3 + 16x^4
From _Philippe Deléham_, Mar 04 2012: (Start)
Triangle (1, 0, -1, 1, 0, 0, 0...) DELTA (0, 2, 0, -1, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  2,  4,  0;
  1,  2,  6,  8,  0;
  1,  2,  8, 14, 16,  0;
  1,  2, 10, 20, 34, 32,  0; (End)

Crossrefs

Cf. A208756, 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*v[n - 1, 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[%]    (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208756 *)

Formula

u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = x*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 04 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 1, T(2,1) = 1 and T(n,k) = 0 if k < 0 or if k > n. (End)
G.f.: -(1+x*y)*x*y/(-1+x*y-x^2*y+2*x^2*y^2+x). - R. J. Mathar, Aug 11 2015

A208756 Triangle of coefficients of polynomials v(n,x) jointly generated with A208755; see the Formula section.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 3, 8, 0, 1, 3, 9, 16, 0, 1, 3, 11, 23, 32, 0, 1, 3, 13, 31, 57, 64, 0, 1, 3, 15, 39, 87, 135, 128, 0, 1, 3, 17, 47, 121, 227, 313, 256, 0, 1, 3, 19, 55, 159, 339, 579, 711, 512, 0, 1, 3, 21, 63, 201, 471, 933, 1431, 1593, 1024, 0, 1, 3, 23, 71
Offset: 1

Views

Author

Clark Kimberling, Mar 01 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0<=k<=n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 02 2012

Examples

			First five rows:
1
0...2
0...1...4
0...1...3...8
0...1...3...9...16
First five polynomials v(n,x):
1
2x
x + 4x^2
x + 3x^2 + 8x^3
x + 3x^2 + 9x^3 + 16^4

Crossrefs

Cf. A208755, 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*v[n - 1, 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[%]    (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208756 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=x*u(n-1,x)+x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
As triangle with 0<=k<=n : G.f.: (1-x+y*x)/(1-(1+y)*x-(2*y^2-y)*x^2). - Philippe Deléham, Mar 02 2012
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 02 2012

A208757 Triangle of coefficients of polynomials u(n,x) jointly generated with A208758; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 16, 1, 2, 10, 24, 44, 1, 2, 12, 32, 76, 120, 1, 2, 14, 40, 112, 232, 328, 1, 2, 16, 48, 152, 368, 704, 896, 1, 2, 18, 56, 196, 528, 1200, 2112, 2448, 1, 2, 20, 64, 244, 712, 1824, 3840, 6288, 6688, 1, 2, 22, 72, 296, 920, 2584, 6144
Offset: 1

Views

Author

Clark Kimberling, Mar 02 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

Examples

			First five rows:
  1;
  1, 2;
  1, 2,  6;
  1, 2,  8, 16;
  1, 2, 10, 24, 44;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 2x +  6x^2
  1 + 2x +  8x^2 + 16x^3
  1 + 2x + 10x^2 + 24x^3 + 44x^4
From _Philippe Deléham_, Mar 18 2012: (Start)
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, ...) begins:
  1
  1, 0
  1, 2,  0
  1, 2,  6,  0
  1, 2,  8, 16,   0
  1, 2, 10, 24,  44,   0
  1, 2, 12, 32,  76, 120,   0
  1, 2, 14, 40, 112, 232, 328, 0; (End)

Crossrefs

Cf. A208758, 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] + 2 x*v[n - 1, 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[%]  (* A208757 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208758 *)

Formula

u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 18 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-2*y*x+2*y*x^2-2*y^2*x^2)/(1-x-2*y*x+2*y*x^2-2*y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) -2*T(n-2,k-1) + 2*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)

A208930 Triangle of coefficients of polynomials v(n,x) jointly generated with A208930; see the Formula section.

Original entry on oeis.org

1, 2, 3, 2, 7, 8, 2, 11, 26, 22, 2, 15, 52, 88, 60, 2, 19, 86, 214, 288, 164, 2, 23, 128, 416, 820, 916, 448, 2, 27, 178, 710, 1824, 2984, 2856, 1224, 2, 31, 236, 1112, 3500, 7464, 10464, 8768, 3344, 2, 35, 302, 1638, 6080, 15884, 29040, 35664, 26592
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
2...7....8
2...11...26...22
2...15...52...88...60
First five polynomials v(n,x):
1
2 + 3x
2 + 7x + 8x^2
2 + 11x + 26x^2 + 22x^3
2 + 15x + 52x^2 + 88x^3 + 60x^4

Crossrefs

Cf. A208909, 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] + 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[%]    (* A208909 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208930 *)

Formula

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

A209137 Triangle of coefficients of polynomials u(n,x) jointly generated with A209138; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 10, 9, 3, 8, 22, 28, 18, 5, 13, 45, 74, 68, 35, 8, 21, 88, 177, 210, 154, 66, 13, 34, 167, 397, 574, 541, 331, 122, 21, 55, 310, 850, 1446, 1656, 1302, 686, 222, 34, 89, 566, 1758, 3434, 4614, 4404, 2982, 1382, 399, 55, 144, 1020
Offset: 1

Views

Author

Clark Kimberling, Mar 05 2012

Keywords

Comments

Every row begins and ends with a Fibonacci number (A000045).
u(n,1) = n-th row sum = 3^(n-1).
Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012
Mirror image of triangle in A209138. - Philippe Deléham, Apr 11 2012

Examples

			First five rows:
  1;
  2,  1;
  3,  4,  2;
  5, 10,  9,  3;
  8, 22, 28, 18, 5;
First three polynomials u(n,x):
  1
  2 +  x
  3 + 4x + 2x^2
From _Philippe Deléham_, Apr 11 2012: (Start)
(1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  4,  2,  0;
   5, 10,  9,  3,  0;
   8, 22, 28, 18,  5,  0;
  13, 45, 74, 68, 35,  8,  0; (End)

Crossrefs

Cf. A209138, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, 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[%]    (* A209137 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209138 *)

Formula

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Apr 11 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-y*x-y*x^2-y^2*x^2)/(1-x-y*x-x^2-y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

A209146 Triangle of coefficients of polynomials u(n,x) jointly generated with A209147; see the Formula section.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 7, 13, 11, 4, 12, 29, 34, 24, 8, 20, 60, 90, 85, 52, 16, 33, 118, 215, 255, 206, 112, 32, 54, 225, 481, 680, 683, 488, 240, 64, 88, 419, 1028, 1682, 1994, 1760, 1136, 512, 128, 143, 767, 2122, 3937, 5361, 5553, 4408, 2608, 1088, 256
Offset: 1

Views

Author

Clark Kimberling, Mar 06 2012

Keywords

Comments

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

Examples

			First five rows:
1
2...1
4...5...2
7...13...11...4
12...29...34...24...8
First three polynomials v(n,x): 1, 2 + x, 4 + 5x + 2x^2.

Crossrefs

Cf. A209146, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, 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[%]    (* A209146 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209147 *)

Formula

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

A209559 Triangle of coefficients of polynomials u(n,x) jointly generated with A209560; see the Formula section.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 5, 8, 3, 1, 9, 17, 15, 4, 1, 15, 38, 39, 24, 5, 1, 25, 76, 104, 74, 35, 6, 1, 41, 149, 242, 229, 125, 48, 7, 1, 67, 282, 543, 607, 440, 195, 63, 8, 1, 109, 524, 1159, 1531, 1308, 769, 287, 80, 9, 1, 177, 957, 2401, 3631, 3660, 2533, 1253, 404
Offset: 1

Views

Author

Clark Kimberling, Mar 10 2012

Keywords

Comments

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

Examples

			First five rows:
1
1...1
3...2....1
5...8....3....1
9...17...15...4...1
First three polynomials v(n,x): 1, 1 + x, 3 + 2x + x^2.

Crossrefs

Cf. A209560, 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];
v[n_, 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[%]   (* A209559 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209560 *)

Formula

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

A209696 Triangle of coefficients of polynomials v(n,x) jointly generated with A209695; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 6, 7, 1, 9, 23, 17, 1, 12, 48, 76, 41, 1, 15, 82, 204, 233, 99, 1, 18, 125, 428, 765, 682, 239, 1, 21, 177, 775, 1907, 2649, 1935, 577, 1, 24, 238, 1272, 4010, 7656, 8680, 5368, 1393, 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363
Offset: 1

Views

Author

Clark Kimberling, Mar 13 2012

Keywords

Comments

Alternating row sums: 1,-2,2,-2,2,-2,2,-2,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012
Mirror image of triangle in A054458. - Philippe Deléham, Mar 24 2012

Examples

			First five rows:
  1;
  1,  3;
  1,  6,  7;
  1,  9, 23, 17;
  1, 12, 48, 76, 41;
First three polynomials v(n,x):
  1
  1 + 3x
  1 + 6x + 7x^2.
From _Philippe Deléham_, Mar 24 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, ...) begins:
  1;
  1,  0;
  1,  3,  0;
  1,  6,  7,  0;
  1,  9, 23, 17,  0;
  1, 12, 48, 76, 41,  0; (End)

Crossrefs

Cf. A054458, A209695, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, 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[%]    (* A209695 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209696 *)

Formula

u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 24 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-2*y*x-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) = 3 and T(n,k) = 0 if k < 0 or if k > n. (End)

A209705 Triangle of coefficients of polynomials u(n,x) jointly generated with A209706; see the Formula section.

Original entry on oeis.org

1, 0, 2, 0, 3, 4, 0, 4, 10, 8, 0, 5, 18, 28, 16, 0, 6, 28, 64, 72, 32, 0, 7, 40, 120, 200, 176, 64, 0, 8, 54, 200, 440, 576, 416, 128, 0, 9, 70, 308, 840, 1456, 1568, 960, 256, 0, 10, 88, 448, 1456, 3136, 4480, 4096, 2176, 512, 0, 11, 108, 624, 2352, 6048
Offset: 1

Views

Author

Clark Kimberling, Mar 12 2012

Keywords

Comments

Alternating row sums: 1,-2,1,-2,1,-2,1,-2,...
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
0...2
0...3...4
0...4...10...8
0...5...18...28...16
First three polynomials v(n,x): 1, 2x, 3x + 4x^2.

Crossrefs

Cf. A209706, 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];
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[%]  (* A209705 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A209706 *)

Formula

u(n,x) = x*u(n-1,x)+x*v(n-1,x),
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.
T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k)-2*T(n-2,k-1), T(1,0)=1, T(2,0)=0, T(2,1)=2, T(3,0)=0, T(3,1)=3, T(3,2)=4, T(n,k)=0 if k<0 or if k>=n. - Philippe Deléham, Dec 27 2013

A209773 Triangle of coefficients of polynomials u(n,x) jointly generated with A209774; see the Formula section.

Original entry on oeis.org

1, 1, 2, 2, 6, 5, 2, 11, 21, 13, 3, 17, 48, 67, 34, 3, 25, 92, 188, 206, 89, 4, 33, 154, 422, 684, 619, 233, 4, 44, 238, 809, 1756, 2365, 1829, 610, 5, 54, 348, 1411, 3801, 6833, 7882, 5334, 1597, 5, 68, 484, 2285, 7369, 16471, 25302, 25549, 15393
Offset: 1

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Author

Clark Kimberling, Mar 15 2012

Keywords

Comments

Last term in row n: F(2n+1), where F=A000045, the Fibonacci numbers
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...2
2...6....5
2...11...21...13
3...17...48...67...34
First three polynomials u(n,x): 1, 1 + 2x, 2 + 6x + 5x^2.

Crossrefs

Cf. A209774, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A209773 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209774 *)

Formula

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