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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A208761 Triangle of coefficients of polynomials u(n,x) jointly generated with A208762; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 1, 12, 18, 8, 1, 20, 52, 50, 16, 1, 30, 120, 186, 126, 32, 1, 42, 240, 534, 576, 306, 64, 1, 56, 434, 1302, 1986, 1654, 718, 128, 1, 72, 728, 2828, 5712, 6632, 4484, 1650, 256, 1, 90, 1152, 5628, 14436, 21912, 20508, 11682, 3726, 512
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2012

Keywords

Comments

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

Examples

			First five rows:
  1;
  1,  2;
  1,  6,  4;
  1, 12, 18,  8;
  1, 20, 52, 50, 16;
First five polynomials u(n,x):
  1
  1 +  2x
  1 +  6x +  4x^2
  1 + 12x + 18x^2 +  8x^3
  1 + 20x + 52x^2 + 50x^3 + 16x^4
From _Philippe Deléham_, Mar 04 2012: (Start)
Triangle (1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   6,   4,   0;
  1,  12,  18,   8,   0;
  1,  20,  52,  50,  16,   0;
  1,  30, 120, 186, 126,  32,  0; (End)

Crossrefs

Cf. A208762, 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] + (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[%]    (* A208761 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208762 *)

Formula

u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
Recurrence: T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 04 2012
G.f.: (-1-x*y+x)*x*y/(-1+x*y+2*x+2*x^2*y^2+x^2*y-x^2). - R. J. Mathar, Aug 12 2015

A208762 Triangle of coefficients of polynomials v(n,x) jointly generated with A208761; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 7, 4, 4, 17, 21, 8, 5, 34, 68, 55, 16, 6, 60, 174, 225, 137, 32, 7, 97, 384, 705, 674, 327, 64, 8, 147, 763, 1863, 2489, 1883, 761, 128, 9, 212, 1400, 4362, 7640, 8012, 5016, 1735, 256, 10, 294, 2412, 9318, 20542, 27996, 24144, 12885, 3897
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2012

Keywords

Comments

Alternating row sums: 1,0,0,0,0,0,0,0,0,... For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0<=k<=n, it is (2, -1/2, 1/2, 0 0 0 0 0 0 0 ...) DELTA (2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012
Row sums are in A055099. - Philippe Deléham, Mar 04 2012

Examples

			First five rows:
1
2...2
3...7....4
4...17...21...8
5...34...68...55...16
First five polynomials v(n,x):
1
2 + 2x
3 + 7x + 4x^2
4 + 17x + 22x^2 + 8x^3
5 + 34x + 68x^2 + 55x^3 + 16x^4

Crossrefs

Cf. A208761, 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] + (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[%]    (* A208761 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208762 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
As triangle T(n,k) with 0<=k<=n : T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Mar 04 2012
G.f.: (-1-x*y)*x*y/(-1+x*y+x^2*y+2*x^2*y^2+2*x-x^2). - R. J. Mathar, Aug 12 2015

A208763 Triangle of coefficients of polynomials u(n,x) jointly generated with A208764; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 10, 14, 1, 2, 14, 26, 38, 1, 2, 18, 38, 90, 94, 1, 2, 22, 50, 158, 250, 246, 1, 2, 26, 62, 242, 470, 762, 622, 1, 2, 30, 74, 342, 754, 1614, 2138, 1606, 1, 2, 34, 86, 458, 1102, 2866, 4870, 6170, 4094, 1, 2, 38, 98, 590, 1514, 4582
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 given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012

Examples

			First five rows:
  1;
  1, 2;
  1, 2,  6;
  1, 2, 10, 14;
  1, 2, 14, 26, 38;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 2x +  6x^2
  1 + 2x + 10x^2 + 14x^3
  1 + 2x + 14x^2 + 26x^3 + 38x^4
From _Philippe Deléham_, Mar 19 2012: (Start)
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  2,  6,  0;
  1,  2, 10, 14,  0;
  1,  2, 14, 26, 38,  0;
  1,  2, 18, 38, 90, 94,  0; (End)

Crossrefs

Cf. A208764, 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];
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[%]    (* A208763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208764 *)

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),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 19 2012: (Start)
G.f.: (1-y*x+2*y*x^2-4*y^2*x^2)/(1-x-y*x+y*x^2-4*y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k >= n. (End)

A208764 Triangle of coefficients of polynomials v(n,x) jointly generated with A208763; see the Formula section.

Original entry on oeis.org

1, 0, 3, 0, 2, 7, 0, 2, 6, 19, 0, 2, 6, 26, 47, 0, 2, 6, 34, 78, 123, 0, 2, 6, 42, 110, 258, 311, 0, 2, 6, 50, 142, 426, 758, 803, 0, 2, 6, 58, 174, 626, 1366, 2282, 2047, 0, 2, 6, 66, 206, 858, 2134, 4594, 6558, 5259, 0, 2, 6, 74, 238, 1122, 3062, 7866, 14334
Offset: 1

Views

Author

Clark Kimberling, Mar 02 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, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -4/3, 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...3
0...2...7
0...2...6...19
0...2...6...26...47
First five polynomials v(n,x):
1
3x
2x + 7x^2
2x + 6x^2 + 19x^3
2x + 6x^2 + 26x^3 + 47x^4

Crossrefs

Cf. A208763, 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];
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[%]    (* A208763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208764 *)

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),
where u(1,x)=1, v(1,x)=1.
As triangle T(n,k), 0 <=k<=n :
G.f.: (1-x+2y*x)/(1-(1+y)*x -(4*y^2-y)*x^2). - Philippe Deléham, Mar 02 2012
As triangle T(n,k), 0<=k<=n : T(n,k) = T(n-1,k) + T(n-1,k-1) + 4*T(n-2,k-2) - T(n-2,k-1) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 02 2012

A208765 Triangle of coefficients of polynomials u(n,x) jointly generated with A208766; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 14, 1, 8, 36, 56, 38, 1, 10, 60, 140, 190, 94, 1, 12, 90, 280, 570, 564, 246, 1, 14, 126, 490, 1330, 1974, 1722, 622, 1, 16, 168, 784, 2660, 5264, 6888, 4976, 1606, 1, 18, 216, 1176, 4788, 11844, 20664, 22392, 14454, 4094, 1
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 given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 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,  4,  6;
  1,  6, 18, 14;
  1,  8, 36, 56, 38;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 4x + 6x^2
  1 + 6x + 18x^2 + 14x^3
  1 + 8x + 36x^2 + 56x^3 + 38x^4
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  4,  6,   0;
  1,  6, 18,  14,   0;
  1,  8, 36,  56,  38,  0;
  1, 10, 60, 140, 190, 94, 0. - _Philippe Deléham_, Mar 18 2012

Links

Crossrefs

Cf. A208766, 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];
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[%]     (* A208765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A208766 *)
Rest[CoefficientList[CoefficientList[Series[(1-x-y*x+2*y*x^2-4*y^2*x^2)/( 1-2*x-y*x+x^2+y*x^2-4*y^2*x^2), {x,0,20}, {y,0,20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)

Formula

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),
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-x-y*x+2*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*y^2*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 4*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.
T(n,k) = binomial(n-1,k)*A026597(k). (End)

A208766 Triangle of coefficients of polynomials v(n,x) jointly generated with A208765; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 6, 7, 1, 9, 21, 19, 1, 12, 42, 76, 47, 1, 15, 70, 190, 235, 123, 1, 18, 105, 380, 705, 738, 311, 1, 21, 147, 665, 1645, 2583, 2177, 803, 1, 24, 196, 1064, 3290, 6888, 8708, 6424, 2047, 1, 27, 252, 1596, 5922, 15498, 26124, 28908, 18423
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 given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -4/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 20 2012

Examples

			First five rows:
  1;
  1,  3;
  1,  6,  7;
  1,  9, 21, 19;
  1, 12, 42, 76, 47;
First five polynomials v(n,x):
  1
  1 +  3x
  1 +  6x +  7x^2
  1 +  9x + 21x^2 + 19x^3
  1 + 12x + 42x^2 + 76x^3 + 47x^4
From _Philippe Deléham_, Mar 20 2012: (Start)
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 3, -2/3, -4/3, 0, 0, ...) begins:
  1;
  1,  0;
  1,  3,  0;
  1,  6,  7,  0;
  1,  9, 21, 19,  0;
  1, 12, 42, 76, 47,  0; (End)

Crossrefs

Cf. A208765, 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];
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[%]     (* A208765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A208766 *)

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),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 20 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-x-y*x+3*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*y^2*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 4*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)

A208905 Triangle of coefficients of polynomials u(n,x) jointly generated with A208906; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 12, 6, 4, 1, 20, 12, 20, 4, 1, 30, 20, 60, 20, 8, 1, 42, 30, 140, 60, 56, 8, 1, 56, 42, 280, 140, 224, 56, 16, 1, 72, 56, 504, 280, 672, 224, 144, 16, 1, 90, 72, 840, 504, 1680, 672, 720, 144, 32, 1, 110, 90, 1320, 840, 3696, 1680, 2640, 720
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....2
1...12...6....4
1...20...12...20...4
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 2x^2
1 + 12x + 6x^2 + 4x^3
1 + 20x + 12x^2 + 20x^3 + 4x^4

Crossrefs

Cf. A208906, 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] + 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[%]  (* A208905 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208906 *)

Formula

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

A208906 Triangle of coefficients of polynomials v(n,x) jointly generated with A208905; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 8, 2, 5, 4, 20, 8, 4, 6, 5, 40, 20, 24, 4, 7, 6, 70, 40, 84, 24, 8, 8, 7, 112, 70, 224, 84, 64, 8, 9, 8, 168, 112, 504, 224, 288, 64, 16, 10, 9, 240, 168, 1008, 504, 960, 288, 160, 16, 11, 10, 330, 240, 1848, 1008, 2640, 960, 880, 160, 32
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...1
3...2...2
4...3...8....2
5...4...20...8...4
First five polynomials v(n,x):
1
2 + x
3 + 2x + 2x^2
4 + 3x + 8x^2 + 2x^3
5 + 4x + 20x^2 + 8x^3 + 4x^4

Crossrefs

Cf. A208905, 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] + 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[%]    (* A208905 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208906 *)

Formula

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

A208908 Triangle of coefficients of polynomials v(n,x) jointly generated with A208923; see the Formula section.

Original entry on oeis.org

1, 2, 2, 2, 5, 4, 2, 9, 15, 8, 2, 13, 33, 37, 16, 2, 17, 59, 103, 91, 32, 2, 21, 93, 221, 297, 213, 64, 2, 25, 135, 407, 739, 807, 491, 128, 2, 29, 185, 677, 1553, 2285, 2105, 1109, 256, 2, 33, 243, 1047, 2907, 5391, 6675, 5319, 2475, 512, 2, 37, 309, 1533
Offset: 1

Views

Author

Clark Kimberling, Mar 04 2012

Keywords

Comments

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,...
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...2
2...5....4
2...9...15...8
2...13...33...37...16
First five polynomials v(n,x):
1
2 + 2x
2 + 5x + 4x^2
2 + 9x + 15x^2 + 8x^3
2 + 13x + 33x^2 + 37x^3 + 16x^4

Crossrefs

Cf. A208923, 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] + 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[%]    (* A208923 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208908 *)

Formula

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

A208909 Triangle of coefficients of polynomials u(n,x) jointly generated with A208930; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 10, 20, 16, 1, 14, 42, 68, 44, 1, 18, 72, 172, 220, 120, 1, 22, 110, 344, 648, 696, 328, 1, 26, 156, 600, 1480, 2336, 2160, 896, 1, 30, 210, 956, 2900, 5984, 8128, 6608, 2448, 1, 34, 272, 1428, 5124, 12984, 23056, 27536, 19984
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...10...20...16
1...14...42...68...44
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 6x^2
1 + 10x + 20x^2 + 16x^3
1 + 14x + 42x^2 + 68x^3 + 44x^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.
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