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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A209131 Triangle of coefficients of polynomials u(n,x) jointly generated with A209132; see the Formula section.

Original entry on oeis.org

1, 2, 1, 2, 4, 3, 2, 8, 12, 5, 2, 12, 28, 28, 11, 2, 16, 52, 84, 68, 21, 2, 20, 84, 188, 236, 156, 43, 2, 24, 124, 356, 612, 628, 356, 85, 2, 28, 172, 604, 1324, 1852, 1612, 796, 171, 2, 32, 228, 948, 2532, 4500, 5316, 4020, 1764, 341, 2, 36, 292, 1404
Offset: 1

Views

Author

Clark Kimberling, Mar 05 2012

Keywords

Comments

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, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

Examples

			First five rows:
  1;
  2,  1;
  2,  4,  3;
  2,  8, 12,  5;
  2, 12, 28, 28, 11;
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, -2, 0, 0, ...) begins:
  1;
  1,   0;
  2,   1,   0;
  2,   4,   3,   0;
  2,   8,  12,   5,   0;
  2,  12,  28,  28,  11,   0;
  2,  16,  52,  84,  68,  21,   0;
  2,  20,  84, 188, 236, 156,  43,   0; (End)

Crossrefs

Cf. A209132, 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_] := 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[%]    (* A209131 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209132 *)

Formula

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

A209132 Triangle of coefficients of polynomials v(n,x) jointly generated with A209131; see the Formula section.

Original entry on oeis.org

1, 0, 3, 0, 4, 5, 0, 4, 12, 11, 0, 4, 20, 36, 21, 0, 4, 28, 76, 92, 43, 0, 4, 36, 132, 244, 228, 85, 0, 4, 44, 204, 508, 716, 540, 171, 0, 4, 52, 292, 916, 1732, 1972, 1252, 341, 0, 4, 60, 396, 1500, 3564, 5436, 5196, 2844, 683, 0, 4, 68, 516, 2292, 6564
Offset: 1

Views

Author

Clark Kimberling, Mar 05 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, 4/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 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

Examples

			First five rows:
  1;
  0,  3;
  0,  4,  5;
  0,  4, 12, 11;
  0,  4, 20, 36, 21;
First three polynomials v(n,x):
  1
     3x
     4x + 5x^2.

Crossrefs

Cf. A209131, 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_] := 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[%]    (* A209131 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209132 *)

Formula

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

A209133 Triangle of coefficients of polynomials u(n,x) jointly generated with A209134; see the Formula section.

Original entry on oeis.org

1, 2, 1, 2, 5, 4, 2, 9, 18, 10, 2, 13, 40, 56, 28, 2, 17, 70, 154, 176, 76, 2, 21, 108, 320, 564, 540, 208, 2, 25, 154, 570, 1344, 1976, 1640, 568, 2, 29, 208, 920, 2700, 5304, 6720, 4928, 1552, 2, 33, 270, 1386, 4848, 11844, 20016, 22320, 14688, 4240
Offset: 1

Views

Author

Clark Kimberling, Mar 05 2012

Keywords

Comments

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, 3, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 10 2012

Examples

			First five rows:
  1;
  2,  1;
  2,  5,  4;
  2,  9, 18, 10;
  2, 13, 40, 56, 28;
First three polynomials u(n,x):
  1
  2 + x
  2 + 5x + 4x^2
From _Philippe Deléham_, Apr 10 2012: (Start)
(1, 1, -2, 1, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, ...) begins:
  1;
  1,   0;
  2,   1,   0;
  2,   5,   4,   0;
  2,   9,  18,  10,   0;
  2,  13,  40,  56,  28,   0;
  2,  17,  70, 154, 176,  76,   0;
  2,  21, 108, 320, 564, 540, 208,  0; (End)

Crossrefs

Cf. A209134, 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_] := 2 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[%]    (* A209133 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209134 *)

Formula

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Apr 10 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-2*y*x+x^2-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(2,1) = 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)

A209134 Triangle of coefficients of polynomials v(n,x) jointly generated with A209133; see the Formula section.

Original entry on oeis.org

1, 0, 4, 0, 4, 10, 0, 4, 18, 28, 0, 4, 26, 72, 76, 0, 4, 34, 132, 256, 208, 0, 4, 42, 208, 572, 864, 568, 0, 4, 50, 300, 1056, 2272, 2808, 1552, 0, 4, 58, 408, 1740, 4800, 8496, 8896, 4240, 0, 4, 66, 532, 2656, 8880, 20208, 30432, 27648, 11584, 0, 4, 74
Offset: 1

Views

Author

Clark Kimberling, Mar 05 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, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 10 2012

Examples

			First five rows:
1
0...4
0...4...10
0...4...18...28
0...4...26...72...76
First three polynomials v(n,x): 1, 4x, 4x + 10x^2.

Crossrefs

Cf. A209133, 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_] := 2 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[%]    (* A209133 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209134 *)

Formula

u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=2x*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) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 4 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Apr 10 2012

A209135 Triangle of coefficients of polynomials u(n,x) jointly generated with A209136; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 5, 3, 4, 14, 16, 5, 5, 30, 54, 39, 11, 6, 55, 144, 171, 98, 21, 7, 91, 329, 561, 503, 229, 43, 8, 140, 672, 1534, 1928, 1380, 532, 85, 9, 204, 1260, 3690, 6106, 6084, 3636, 1203, 171, 10, 285, 2208, 8058, 16852, 21890, 18060, 9249, 2694
Offset: 1

Views

Author

Clark Kimberling, Mar 05 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012

Examples

			First five rows:
  1;
  2,  1;
  3,  5,  3;
  4, 14, 16,  5;
  5, 30, 54, 39, 11;
First three polynomials u(n,x):
  1
  2 + x
  3 + 5x + 3x^2
From _Philippe Deléham_, Apr 11 2012: (Start)
(1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  3,  5,  3,  0;
  4, 14, 16,  5,  0;
  5, 30, 54, 39, 11,  0; (End)

Crossrefs

Cf. A209136, 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_] := 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[%]    (* A209135 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209136 *)

Formula

u(n,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, Apr 11 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x-y*x+x^2-y*x^2-2*y^2*x^2)/(1-2*x-y*x+x^2-y*x^2-2*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) + 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. (End)

A209136 Triangle of coefficients of polynomials v(n,x) jointly generated with A209135; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 8, 5, 1, 15, 23, 11, 1, 24, 66, 66, 21, 1, 35, 150, 240, 165, 43, 1, 48, 295, 678, 747, 404, 85, 1, 63, 525, 1631, 2547, 2157, 947, 171, 1, 80, 868, 3500, 7246, 8560, 5864, 2182, 341, 1, 99, 1356, 6888, 18126, 28018, 26592, 15318, 4929, 683
Offset: 1

Views

Author

Clark Kimberling, Mar 05 2012

Keywords

Comments

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 given by (1, 0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012

Examples

			First five rows:
  1;
  1,  3;
  1,  8,  5;
  1, 15, 23, 11;
  1, 24, 66, 66, 21;
First three polynomials v(n,x):
  1
  1 + 3x
  1 + 8x + 5x^2.
From _Philippe Deléham_, Apr 11 2012: (Start)
(1, 0, 2/3, 1/3, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   3,   0;
  1,   8,   5,   0;
  1,  15,  23,  11,   0;
  1,  24,  66,  66,  21,   0;
  1,  35, 150, 240, 165,  43,   0; (End)

Crossrefs

Cf. A209135, 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_] := 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[%]    (* A209135 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209136 *)

Formula

u(n,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, Apr 11 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x-y*x+y*x^2-2*y^2*x^2)/(1-2*x-y*x+x^2-y*x^2-2*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) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

A209139 Triangle of coefficients of polynomials u(n,x) jointly generated with A209140; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 5, 3, 5, 12, 15, 7, 8, 27, 45, 42, 17, 13, 55, 119, 151, 116, 41, 21, 108, 282, 458, 480, 315, 99, 34, 205, 630, 1228, 1631, 1467, 845, 239, 55, 381, 1343, 3054, 4849, 5502, 4358, 2244, 577, 89, 696, 2769, 7173, 13218, 17895, 17838, 12666
Offset: 1

Views

Author

Clark Kimberling, Mar 05 2012

Keywords

Comments

Column 1: A000045 (Fibonacci numbers).
Alternating row sums: 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 given by (1, 1, -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, Apr 11 2012

Examples

			First five rows:
  1;
  2,  1;
  3,  5,  3;
  5, 12, 15,  7;
  8, 27, 45, 42, 17;
First three polynomials u(n,x):
  1
  2 + x
  3 + 5x + 3x^2
From _Philippe Deléham_, Apr 11 2012: (Start)
(1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,   0;
   3,  5,   3,   0;
   5, 12,  15,   7,   0;
   8, 27,  45,  42,  17,  0;
  13, 55, 119, 151, 116, 41, 0; (End)

Crossrefs

Cf. A209140, 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] + 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[%]    (* A209139 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209140 *)

Formula

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

A209140 Triangle of coefficients of polynomials v(n,x) jointly generated with A209139; see the Formula section.

Original entry on oeis.org

1, 1, 3, 2, 5, 7, 3, 12, 18, 17, 5, 23, 51, 58, 41, 8, 45, 118, 189, 175, 99, 13, 84, 264, 506, 645, 507, 239, 21, 155, 558, 1268, 1950, 2085, 1428, 577, 34, 281, 1145, 2974, 5395, 6998, 6482, 3940, 1393, 55, 504, 2286, 6687, 13851, 21141, 23856
Offset: 1

Views

Author

Clark Kimberling, Mar 05 2012

Keywords

Comments

Column 1: Fibonacci numbers, A000045.
Alternating row sums: (-2)^(n-1).
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
  1;
  1,  3;
  2,  5,  7;
  3, 12, 18, 17;
  5, 23, 51, 58, 41;
First three polynomials v(n,x):
  1
  1 + 3x
  2 + 5x + 7x^2.

Crossrefs

Cf. A209139, 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] + 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[%]    (* A209139 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209140 *)

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.
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 3, T(n,k) = 0 if k < 0 or if k >= n. - Philippe Deléham, Apr 11 2012

A209141 Triangle of coefficients of polynomials u(n,x) jointly generated with A209142; see the Formula section.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 8, 16, 12, 3, 16, 44, 49, 25, 5, 32, 112, 166, 127, 50, 8, 64, 272, 504, 513, 301, 96, 13, 128, 640, 1424, 1808, 1408, 670, 180, 21, 256, 1472, 3824, 5816, 5641, 3562, 1427, 331, 34, 512, 3328, 9888, 17520, 20330, 15981, 8494, 2939
Offset: 1

Views

Author

Clark Kimberling, Mar 06 2012

Keywords

Comments

Each row begins with a power of 2 and ends with a Fibonacci number. Alternating row sums: all 1's. For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle T(n,k) given by (1, 1, 0, 0, 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 07 2012

Examples

			First five rows:
1
2....1
4....5....2
8....16...12...3
16...44...49...25...5
First three polynomials u(n,x):  1, 2 + x, 4 + 5x + 2x^2
Triangle (1, 1, 0, 0, 0...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
4, 5, 2, 0
8, 16, 12, 3, 0
16, 44, 49, 25, 5, 0
32, 112, 166, 127, 50, 8, 0

Crossrefs

Cf. A209142, 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 + 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[%]    (* A209141 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209142 *)

Formula

u(n,x)=u(n-1,x)+(x+1)*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.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Mar 07 2012
G.f.: -x*y/(-1+x*y+x^2*y^2+2*x+x^2*y). - R. J. Mathar, Aug 12 2015

A209142 Triangle of coefficients of polynomials v(n,x) jointly generated with A209141; see the Formula section.

Original entry on oeis.org

1, 2, 2, 4, 7, 3, 8, 20, 17, 5, 16, 52, 65, 37, 8, 32, 128, 210, 176, 75, 13, 64, 304, 616, 679, 428, 146, 21, 128, 704, 1696, 2312, 1921, 971, 276, 34, 256, 1600, 4464, 7240, 7449, 4970, 2097, 511, 55, 512, 3584, 11360, 21344, 26146, 21622, 12056
Offset: 1

Views

Author

Clark Kimberling, Mar 06 2012

Keywords

Comments

Each row begins with a power of 2 and ends with a Fibonacci number.
Alternating row sums: 1,0,0,0,0,0,0,0,0,0,0,...
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0<=k<=n, it is (2, 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 07 2012

Examples

			First five rows:
1
2....2
4....7....3
8....20...17...5
16...52...65...37...8
First three polynomials v(n,x): 1, 2 + 2x, 4 + 7x + 3x^2.

Crossrefs

Cf. A209141, 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 + 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[%]    (* A209141 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209142 *)

Formula

u(n,x)=u(n-1,x)+(x+1)*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), 0<=k<=n : T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 07 2012
G.f.: (-1-x*y)*x*y/(-1+x*y+x^2*y^2+2*x+x^2*y). - R. J. Mathar, Aug 12 2015
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