A208510 -id:A208510 - cp's OEIS frontend

A208660 Triangle of coefficients of polynomials u(n,x) jointly generated with A208904; see the Formula section.

1, 1, 2, 1, 8, 2, 1, 18, 14, 2, 1, 32, 52, 20, 2, 1, 50, 140, 104, 26, 2, 1, 72, 310, 380, 174, 32, 2, 1, 98, 602, 1106, 806, 262, 38, 2, 1, 128, 1064, 2744, 2924, 1472, 368, 44, 2, 1, 162, 1752, 6048, 8892, 6412, 2432, 492, 50, 2, 1, 200, 2730, 12168, 23652

Offset: 1

Clark Kimberling, Mar 03 2012

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

			First five rows:
1
1...2
1...8....2
1...18...14...2
1...32...52...20...2
First five polynomials u(n,x):
1
1 + 2x
1 + 8x + 2x^2
1 + 18x + 14x^2 + 2x^3
1 + 32x + 52x^2 + 20x^3 + 2x^4

Cf. A208904, A208510.

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_] := 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[%]    (* A208660 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208904 *)

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

A208749 Triangle of coefficients of polynomials u(n,x) jointly generated with A208750; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 12, 10, 4, 1, 20, 32, 24, 4, 1, 30, 80, 88, 36, 8, 1, 42, 170, 256, 180, 72, 8, 1, 56, 322, 644, 660, 384, 104, 16, 1, 72, 560, 1456, 1992, 1568, 704, 192, 16, 1, 90, 912, 3024, 5256, 5360, 3392, 1344, 272, 32, 1, 110, 1410, 5856, 12552

Offset: 1

Clark Kimberling, Mar 02 2012

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

Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

			First five rows:
  1;
  1,  2;
  1,  6,  2;
  1, 12, 10,  4;
  1, 20, 32, 24,  4;
First five polynomials u(n,x):
  1
  1 +  2x
  1 +  6x +  2x^2
  1 + 12x + 10x^2 +  4x^3
  1 + 20x + 32x^2 + 24x^3 + 4x^4
From _Philippe Deléham_, Mar 14 2012: (Start)
(1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  6,  2,  0;
  1, 12, 10,  4,  0;
  1, 20, 32, 24,  4,  0;
  1, 30, 80, 88, 36,  8,  0; (End)

Cf. A208750, A208510.

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];
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[%]    (* A208749 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208750 *)

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),
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle: g.f.: (1-x-2*y^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - Philippe Deléham, Mar 14 2012
Recurrence: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > =n. - Philippe Deléham, Mar 14 2012

A208750 Triangle of coefficients of polynomials v(n,x) jointly generated with A208749; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 11, 10, 2, 5, 24, 32, 16, 4, 6, 45, 84, 72, 32, 4, 7, 76, 194, 240, 156, 48, 8, 8, 119, 406, 666, 592, 300, 88, 8, 9, 176, 784, 1632, 1896, 1344, 576, 128, 16, 10, 249, 1416, 3648, 5344, 4904, 2848, 1024, 224, 16, 11, 340, 2418, 7584

Offset: 1

Clark Kimberling, Mar 02 2012

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

Subtriangle of the triangle T(n,k) given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 16 2012

			First five rows:
  1;
  2,  1;
  3,  4,  2;
  4, 11, 10,  2;
  5, 24, 32, 16,  4;
First five polynomials v(n,x):
  1
  2 +   x
  3 +  4x +  2x^2
  4 + 11x + 10x^2 +  2x^3
  5 + 24x + 32x^2 + 16x^3 + 4x^4
From _Philippe Deléham_, Mar 16 2012: (Start)
(1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -2, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  3,  4,  2,  0;
  4, 11, 10,  2,  0;
  5, 24, 32, 16,  4,  0;
  6, 45, 84, 72, 32,  4,  0; (End)

Cf. A208749, A208510.

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];
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[%]    (* A208749 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208750 *)

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),
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle with 0 <= k <= n: g.f.: (1-x+x^2-y*x^2-2*t^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - Philippe Deléham, Mar 16 2012
As DELTA-triangle: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1) + 2*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. - Philippe Deléham, Mar 16 2012

A208751 Triangle of coefficients of polynomials u(n,x) jointly generated with A208752; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 40, 18, 2, 1, 30, 100, 86, 24, 2, 1, 42, 210, 294, 150, 30, 2, 1, 56, 392, 812, 656, 232, 36, 2, 1, 72, 672, 1932, 2268, 1240, 332, 42, 2, 1, 90, 1080, 4116, 6624, 5172, 2100, 450, 48, 2, 1, 110, 1650, 8052, 17028, 17996

Offset: 1

Clark Kimberling, Mar 01 2012

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

Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 17 2012

			First five rows:
  1;
  1,  2;
  1,  6,  2;
  1, 12, 12,  2;
  1, 20, 40, 18,  2;
First five polynomials u(n,x):
  1
  1 +  2x
  1 +  6x +  2x^2
  1 + 12x + 12x^2 +  2x^3
  1 + 20x + 40x^2 + 18x^3 + 2x^4
From _Philippe Deléham_, Mar 17 2012: (Start)
(1, 0, 1, 0, 0, ...) DELTA (0, 2, -1, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   6,   2,   0;
  1,  12,  12,   2,   0;
  1,  20,  40,  18,   2,   0;
  1,  30, 100,  86,  24,   2,   0; (End)

Cf. A208752, A208510.

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_] := 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[%]    (* A208751 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208752 *)

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.
From Philippe Deléham, Mar 17 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-x-y*x)/(1-2*x-y*x+x^2-y*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1), 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)

A208752 Triangle of coefficients of polynomials v(n,x) jointly generated with A208751; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 14, 8, 1, 5, 30, 34, 11, 1, 6, 55, 104, 63, 14, 1, 7, 91, 259, 253, 101, 17, 1, 8, 140, 560, 806, 504, 148, 20, 1, 9, 204, 1092, 2178, 1966, 884, 204, 23, 1, 10, 285, 1968, 5202, 6412, 4090, 1420, 269, 26, 1, 11, 385, 3333, 11286, 18238

Offset: 1

Clark Kimberling, Mar 01 2012

For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 17 2012
Setting v(0,x) = 0, the sequence of polynomials {v(n,x) : n >= 0} satisfies the second-order recurrence v(n,x) = (x + 2)*v(n-1,x) + (x - 1)*v(n-2,x) with v(0,x) = 0 and v(1,x) = 1. Then by Norfleet, this sequence of polynomials is a strong divisibility sequence of polynomials in the ring Z[x], that is gcd(v(n,x), v(m,x)) = v(gcd(n,m),x). In particular, if n divides m then v(n,x) divides v(m,x) in Z[x]. - Peter Bala, Feb 07 2024

			First five rows:
  1
  2   1
  3   5    1
  4   14   8    1
  5   30   34   11   1
First five polynomials u(n,x) - see A208751:
  1
  1 + 2*x
  1 + 6*x + 2*x^2
  1 + 12*x + 12*x^2 + 2*x^3
  1 + 20*x + 40*x^2 + 18*x^3 + 2*x^4
(0, 2, -1/2, 1/2, 0, 0, ...) DELTA (1, 0, 1/2, -1/2, 0, 0, ...) begins:
  1
  0, 1
  0, 2, 1
  0, 3, 5, 1
  0, 4, 14, 8, 1
  0, 5, 30, 34, 11, 1. - _Philippe Deléham_, Mar 17 2012

M. Norfleet, Characterization of second-order strong divisibility sequences of polynomials, The Fibonacci Quarterly, 43(2) (2005), 166-169.

Cf. A208751, A208510.

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_] := 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[%]    (* A208751 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208752 *)

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1), T(1,0) = 1, T(2,0) = 2, T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 17 2012
G.f.: -x*y/(-1+2*x-x^2+x^2*y+x*y). - R. J. Mathar, Aug 12 2015

A208753 Triangle of coefficients of polynomials u(n,x) jointly generated with A208754; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 8, 1, 18, 4, 1, 32, 24, 1, 50, 80, 8, 1, 72, 200, 64, 1, 98, 420, 280, 16, 1, 128, 784, 896, 160, 1, 162, 1344, 2352, 864, 32, 1, 200, 2160, 5376, 3360, 384, 1, 242, 3300, 11088, 10560, 2464, 64, 1, 288, 4840, 21120, 28512, 11264, 896, 1

Offset: 1

Clark Kimberling, Mar 03 2012

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

			First five rows:
1
1...2
1...8
1...18...4
1...32...24
First five polynomials u(n,x):
1
1 + 2x
1 + 8x
1 + 18x + 4x^2
1 + 32x + 24x^2

Cf. A208754, A208510.

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_] := 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[%]    (* A208753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208754 *)

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

A208754 Triangle of coefficients of polynomials v(n,x) jointly generated with A208753; see the Formula section.

Original entry on oeis.org

1, 3, 5, 2, 7, 10, 9, 28, 4, 11, 60, 28, 13, 110, 108, 8, 15, 182, 308, 72, 17, 280, 728, 352, 16, 19, 408, 1512, 1248, 176, 21, 570, 2856, 3600, 1040, 32, 23, 770, 5016, 8976, 4400, 416, 25, 1012, 8316, 20064, 14960, 2880, 64, 27, 1300, 13156, 41184

Offset: 1

Clark Kimberling, Mar 03 2012

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

			First five rows:
1
3
5...2
7...10
9...28...4
First five polynomials v(n,x):
1
3
5 + 2x
7 + 10x
9 + 28x + 4x^2

Cf. A208753, A208510.

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_] := 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[%]    (* A208753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208754 *)

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

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

Original entry on oeis.org

1, 0, 3, 0, 1, 8, 0, 1, 4, 22, 0, 1, 4, 16, 60, 0, 1, 4, 18, 56, 164, 0, 1, 4, 20, 68, 188, 448, 0, 1, 4, 22, 80, 248, 608, 1224, 0, 1, 4, 24, 92, 312, 864, 1920, 3344, 0, 1, 4, 26, 104, 380, 1152, 2928, 5952, 9136, 0, 1, 4, 28, 116, 452, 1472, 4128, 9696, 18192

Offset: 1

Clark Kimberling, Mar 02 2012

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

As triangle T(n,k) with 0 <= k <= n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 02 2012

			First five rows:
  1;
  0,  3;
  0,  1,  8;
  0,  1,  4, 22;
  0,  1,  4, 16, 60;
First five polynomials v(n,x):
  1
     3x
      x + 8x^2
      x + 4x^2 + 22x^3
      x + 4x^2 + 16x^3 + 60^x4

Cf. A208757, A208510.

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 *)

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

A208759 Triangle of coefficients of polynomials u(n,x) jointly generated with A208760; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 16, 16, 1, 8, 30, 56, 44, 1, 10, 48, 128, 188, 120, 1, 12, 70, 240, 504, 608, 328, 1, 14, 96, 400, 1080, 1872, 1920, 896, 1, 16, 126, 616, 2020, 4512, 6672, 5952, 2448, 1, 18, 160, 896, 3444, 9352, 17856, 23040, 18192, 6688, 1, 20, 198, 1248, 5488, 17472, 40600, 67776, 77616, 54976, 18272

Offset: 1

Clark Kimberling, Mar 02 2012

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, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

			First five rows:
  1;
  1,  2;
  1,  4,  6;
  1,  6, 16, 16;
  1,  8, 30, 56, 44;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 4x +  6x^2
  1 + 6x + 16x^2 + 16x^3
  1 + 8x + 30x^2 + 56x^3 + 44x^4
From _Philippe Deléham_, Mar 18 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   4,   6,   0;
  1,   6,  16,  16,   0;
  1,   8,  30,  56,  44,   0;
  1,  10,  48, 128, 188, 120,  0; (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A208760, A208510.

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];
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[%]  (* A208759 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208760 *)
Rest[CoefficientList[CoefficientList[Series[(1-2*y*x-2*y^2*x^2)/(1-x-2*y*x- 2*y^2*x^2), {x,0,20}, {y,0,20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)

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),
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.: (1-2y*x-2*y^2*x^2)/(1-x-2*y*x-2*y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,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)

Terms a(58) onward added by G. C. Greubel, Mar 28 2018

A208760 Triangle of coefficients of polynomials v(n,x) jointly generated with A208759; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 5, 8, 1, 7, 20, 22, 1, 9, 36, 72, 60, 1, 11, 56, 158, 244, 164, 1, 13, 80, 288, 632, 796, 448, 1, 15, 108, 470, 1320, 2376, 2528, 1224, 1, 17, 140, 712, 2420, 5592, 8544, 7872, 3344, 1, 19, 176, 1022, 4060, 11372, 22368, 29712, 24144, 9136

Offset: 1

Clark Kimberling, Mar 02 2012

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

Subtriangle of the triangle given by (1, 0, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

			First five rows:
  1;
  1,  3;
  1,  5,  8;
  1,  7, 20, 22;
  1,  9, 36, 72, 60;
First five polynomials v(n,x):
  1
  1 + 3x
  1 + 5x +  8x^2
  1 + 7x + 20x^2 + 22x^3
  1 + 9x + 36x^2 + 72x^3 + 60x^4
From _Philippe Deléham_, Mar 18 2012: (Start)
(1, 0, -1/3, 1/3, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, ...) begins:
  1;
  1,   0;
  1,   3,   0;
  1,   5,   8,   0;
  1,   7,  20,  22,   0;
  1,   9,  36,  72,  60,   0;
  1,  11,  56, 158, 244, 164,  0; (End)

Cf. A208759, A208510.

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];
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[%]  (* A208759 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208760 *)

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),
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+y*x^2-2*y^2*x^2)/(1-x-2*y*x-2*y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,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) = 3 and T(n,k) = 0 if k < 0 or if k > n. (End)

cp's OEIS Frontend

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A208750 Triangle of coefficients of polynomials v(n,x) jointly generated with A208749; see the Formula section.

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A208751 Triangle of coefficients of polynomials u(n,x) jointly generated with A208752; see the Formula section.

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A208752 Triangle of coefficients of polynomials v(n,x) jointly generated with A208751; see the Formula section.

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A208753 Triangle of coefficients of polynomials u(n,x) jointly generated with A208754; see the Formula section.

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A208754 Triangle of coefficients of polynomials v(n,x) jointly generated with A208753; see the Formula section.

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

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