A208510 -id:A208510 - cp's OEIS frontend

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

1, 3, 1, 5, 4, 2, 7, 13, 12, 2, 9, 32, 40, 16, 4, 11, 65, 108, 80, 36, 4, 13, 116, 258, 288, 180, 48, 8, 15, 189, 560, 842, 700, 324, 96, 8, 17, 288, 1120, 2144, 2312, 1536, 640, 128, 16, 19, 417, 2088, 4944, 6728, 5832, 3232, 1088, 240, 16, 21, 580, 3666

Offset: 1

Clark Kimberling, Mar 04 2012

Alternating row sums: 1,2,3,4,5,6,7,8,9,10,...

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

			First five rows:
1
3...1
5...4....2
7...13...12...2
9...32...40...16...4
First five polynomials v(n,x):
1
3 + x
5 + 4x + 2x^2
7 + 13x + 12x^2 + 2x^3
9 + 32x + 40x^2 + 16x^3 + 4x^4

Cf. A208921, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
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;
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 *)

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.

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

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 1, 10, 14, 8, 1, 14, 32, 38, 16, 1, 18, 58, 104, 90, 32, 1, 22, 92, 222, 296, 214, 64, 1, 26, 134, 408, 738, 808, 490, 128, 1, 30, 184, 678, 1552, 2286, 2104, 1110, 256, 1, 34, 242, 1048, 2906, 5392, 6674, 5320, 2474, 512, 1, 38, 308

Offset: 1

Clark Kimberling, Mar 04 2012

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

			First five rows:
1
1...2
1...6....4
1...10...14...8
1...14...32...38...16
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 4x^2
1 + 10x + 14x^2 + 8x^3
1 + 14x + 32x^2 + 38x^3 + 16x^4

Cf. A208908, 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] + 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 *)

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.

A208931 Triangle of coefficients of polynomials u(n,x) jointly generated with A208932; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 8, 4, 1, 18, 20, 8, 1, 32, 64, 56, 16, 1, 50, 160, 224, 136, 32, 1, 72, 340, 680, 664, 328, 64, 1, 98, 644, 1736, 2416, 1872, 760, 128, 1, 128, 1120, 3920, 7264, 7856, 4984, 1736, 256, 1, 162, 1824, 8064, 19056, 26992, 23768, 12832, 3896

Offset: 1

Clark Kimberling, Mar 04 2012

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

			First five rows:
1
1...2
1...8....4
1...18...20...8
1...32...64...56...16
First five polynomials u(n,x):
1
1 + 2x
1 + 8x + 4x^2
1 + 18x + 20x^2 + 8x^3
1 + 32x + 64x^2 + 56x^3 + 16x^4

Cf. A208931, 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] + (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[%]    (* A208931 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208932 *)

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)+1,
where u(1,x)=1, v(1,x)=1.

A208932 Triangle of coefficients of polynomials v(n,x) jointly generated with A208932; see the Formula section.

Original entry on oeis.org

1, 3, 2, 5, 8, 4, 7, 22, 24, 8, 9, 48, 84, 60, 16, 11, 90, 228, 264, 148, 32, 13, 152, 528, 876, 772, 348, 64, 15, 238, 1092, 2424, 2992, 2112, 804, 128, 17, 352, 2072, 5896, 9568, 9392, 5548, 1820, 256, 19, 498, 3672, 13008, 26648, 34080, 27780

Offset: 1

Clark Kimberling, Mar 04 2012

Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,...

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

			First five rows:
1
3...2
5...8....4
7...22...24...8
9...48...84...60...16
First five polynomials v(n,x):
1
3 + 2x
5 + 8x + 4x^2
7 + 22x + 24x^2 + 8x^3
9 + 48x + 84x^2 + 60x^3 + 16x^4

Cf. A208930, 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] + (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[%]    (* A208931 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208932 *)

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)+1,
where u(1,x)=1, v(1,x)=1.

A209125 Triangle of coefficients of polynomials u(n,x) jointly generated with A164975; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 9, 9, 4, 8, 20, 25, 20, 8, 13, 40, 65, 65, 44, 16, 21, 78, 150, 190, 162, 96, 32, 34, 147, 331, 490, 521, 392, 208, 64, 55, 272, 697, 1192, 1473, 1368, 928, 448, 128, 89, 495, 1425, 2745, 3888, 4185, 3480, 2160, 960, 256, 144, 890

Offset: 1

Clark Kimberling, Mar 05 2012

Alternating row sums:  1,1,1,1,1,1,1,1,1,1,1,1,1,...
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, 0, 0, 0, 0, 0, 0, 0, ....) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

			First five rows:
  1;
  2,  1;
  3,  4,  2;
  5,  9,  9,  4;
  8, 20, 25, 20,  8;
First three polynomials u(n,x):
  1
  2 + x
  3 + 4x + 2x^2
From _Philippe Deléham_, Mar 21 2012: (Start)
(1, 1, -1, 0, 0, ...) DELTA (0, 1, 1, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  3,  4,  2,  0;
  5,  9,  9,  4, 0;
  8, 20, 25, 20, 8, 0; (End)

Cf. A164975, A208510.

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];
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[%]    (* A209125 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A164975 *)

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),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 21 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-2*y*x)/(1-x-2*y*x-x^2+y*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2, T(n,k) = 0 f k < 0 or if k > n. (End)

A209126 Triangle of coefficients of polynomials u(n,x) jointly generated with A209127; see the Formula section.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 5, 7, 3, 2, 7, 14, 13, 5, 2, 9, 23, 32, 25, 8, 2, 11, 34, 62, 71, 46, 13, 2, 13, 47, 105, 156, 149, 84, 21, 2, 15, 62, 163, 295, 367, 304, 151, 34, 2, 17, 79, 238, 505, 767, 827, 604, 269, 55, 2, 19, 98, 332, 805, 1435, 1889, 1798, 1177, 475

Offset: 1

Clark Kimberling, Mar 05 2012

u(n,n) = A000045(n), Fibonacci numbers.
Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -2, 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, Mar 21 2012

			First five rows:
  1;
  2,  1;
  2,  3,  2;
  2,  5,  7,  3;
  2,  7, 14, 13,  5;
First three polynomials u(n,x):
  1
  2 + x
  2 + 3x + 2x^2
From _Philippe Deléham_, Mar 21 2012: (Start)
(1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  2,  3,  2,  0;
  2,  5,  7,  3,  0;
  2,  7, 14, 13,  5,  0; (End)

Cf. A209127, A208510.

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*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[%]    (* A209126 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209127 *)

u(n,x) = u(n-1,x) + (x+1)*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 21 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-y*x+x^2-y^2*x^2)/(1-x-y*x-y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)

A209127 Triangle of coefficients of polynomials v(n,x) jointly generated with A209126; see the Formula section.

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 5, 5, 0, 2, 7, 12, 8, 0, 2, 9, 21, 25, 13, 0, 2, 11, 32, 53, 50, 21, 0, 2, 13, 45, 94, 124, 96, 34, 0, 2, 15, 60, 150, 250, 273, 180, 55, 0, 2, 17, 77, 223, 445, 617, 577, 331, 89, 0, 2, 19, 96, 315, 728, 1212, 1444, 1181, 600, 144, 0, 2, 21

Offset: 1

Clark Kimberling, Mar 05 2012

u(n,n)=(1,2,3,5,8,13,21,...)=A000045(n+1), Fibonacci numbers.
Alternating row sums: (1,-2,1,-2,1,-2,1,-2,...
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0<=k<=n, it is (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

			First five rows:
1
0...2
0...2...3
0...2...5...5
0...2...7...12...8
First three polynomials v(n,x): 1, 2x, 2x + 3x^2.

Cf. A209126, A208510.

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*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[%]    (* A209126 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209127 *)

u(n,x)=u(n-1,x)+(x+1)*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.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 21 2012
G.f.: (-1-x*y+x)*x*y/(-1+x*y+x+x^2*y^2). - R. J. Mathar, Aug 12 2015

A209128 Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section.

Original entry on oeis.org

1, 2, 1, 2, 4, 3, 2, 6, 12, 7, 2, 8, 22, 32, 17, 2, 10, 34, 70, 86, 41, 2, 12, 48, 124, 216, 228, 99, 2, 14, 64, 196, 428, 644, 600, 239, 2, 16, 82, 288, 744, 1408, 1876, 1568, 577, 2, 18, 102, 402, 1188, 2664, 4476, 5364, 4074, 1393, 2, 20, 124, 540, 1786

Offset: 1

Clark Kimberling, Mar 05 2012

For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012
Row sums are powers of 3 (A000244). - Philippe Deléham, Mar 21 2012

			First five rows:
  1;
  2,  1;
  2,  4,  3;
  2,  6, 12,  7;
  2,  8, 22, 32, 17;
First three polynomials u(n,x):
  1
  2 + x
  2 + 4x + 3x^2
From _Philippe Deléham_, Mar 21 2012: (Start)
(1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  2,  4,  3,  0;
  2,  6, 12,  7,  0;
  2,  8, 22, 32, 17,  0; (End)

Cf. A209129, A208510.

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*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[%]    (* A209128 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209129 *)

u(n,x) = u(n-1,x) + (x+1)*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 21 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-2*y*x+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-1) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)

A209129 Triangle of coefficients of polynomials v(n,x) jointly generated with A209128; see the Formula section.

Original entry on oeis.org

1, 0, 3, 0, 2, 7, 0, 2, 8, 17, 0, 2, 10, 28, 41, 0, 2, 12, 42, 88, 99, 0, 2, 14, 58, 154, 262, 239, 0, 2, 16, 76, 240, 524, 752, 577, 0, 2, 18, 96, 348, 908, 1692, 2104, 1393, 0, 2, 20, 118, 480, 1440, 3224, 5260, 5776, 3363, 0, 2, 22, 142, 638, 2148, 5544

Offset: 1

Clark Kimberling, Mar 05 2012

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, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

			First five rows:
1
0...3
0...2....7
0...2....8...17
0...2...10...28...41
First three polynomials v(n,x): 1, 3x, 2x + 7x^2.

Cf. A209128, A208510.

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*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[%]    (* A209128 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209129 *)

u(n,x)=u(n-1,x)+(x+1)*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.
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 21 2012
G.f.: (-1+x-x*y)*x*y/(-1+x+2*x*y-x^2*y+x^2*y^2). - R. J. Mathar, Aug 12 2015

A209130 Triangle of coefficients of polynomials v(n,x) jointly generated with A102756; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 9, 12, 5, 1, 14, 31, 27, 8, 1, 20, 65, 89, 55, 13, 1, 27, 120, 230, 222, 108, 21, 1, 35, 203, 511, 684, 514, 205, 34, 1, 44, 322, 1022, 1777, 1834, 1125, 381, 55, 1, 54, 486, 1890, 4095, 5442, 4563, 2367, 696, 89, 1, 65, 705, 3288, 8625

Offset: 1

Clark Kimberling, Mar 05 2012

Top edge:  (1,2,3,5,8,...) = A000045(n+1), Fibonacci numbers.
Alternating row sums: 1,-1,-1,-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 T(n,k) given by (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, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012

			First five rows:
  1;
  1,  2;
  1,  5,  3;
  1,  9, 12,  5;
  1, 14, 31, 27,  8;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 5x + 3x^2.
From _Philippe Deléham_, Mar 08 2012: (Start)
(1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  5,  3,  0;
  1,  9, 12,  5,  0;
  1, 14, 31, 27,  8,  0;
  1, 20, 65, 89, 55, 13, 0; ...
with row sums 1, 1, 3, 9, 27, 81, 243, 729, ... (powers of 3). (End)

Cf. A102756, A208510.

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*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[%]    (* A102756 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209130 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 08 2012: (Start)
As DELTA-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-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-x-y*x+y*x^2-y^2*x^2)/(1-(2+y)*x-(y^2-1)*x^2).
Sum_{k=0..n, n>=1} T(n,k)*x^k = A153881(n), A000012(n), A000244(n-1), A126473(n-1) for x = -1, 0, 1, 2 respectively. (End)

cp's OEIS Frontend

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

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

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A208931 Triangle of coefficients of polynomials u(n,x) jointly generated with A208932; see the Formula section.

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A208932 Triangle of coefficients of polynomials v(n,x) jointly generated with A208932; see the Formula section.

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A209125 Triangle of coefficients of polynomials u(n,x) jointly generated with A164975; see the Formula section.

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A209126 Triangle of coefficients of polynomials u(n,x) jointly generated with A209127; see the Formula section.

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A209127 Triangle of coefficients of polynomials v(n,x) jointly generated with A209126; see the Formula section.

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A209128 Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section.

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