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.

Previous Showing 11-20 of 345 results. Next

A208914 Triangle of coefficients of polynomials v(n,x) jointly generated with A208913; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 6, 16, 8, 5, 8, 40, 32, 16, 6, 10, 80, 80, 96, 32, 7, 12, 140, 160, 336, 192, 64, 8, 14, 224, 280, 896, 672, 512, 128, 9, 16, 336, 448, 2016, 1792, 2304, 1024, 256, 10, 18, 480, 672, 4032, 4032, 7680, 4608, 2560, 512, 11, 20, 660, 960
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...2
3...4...4
4...6...16...8
5...8...40...32...16
First five polynomials v(n,x):
1
2 + 2x
3 + 4x + 4x^2
4 + 6x + 16x^2 + 8x^3
5 + 8x + 40x^2 + 32x^3 + 16x^4

Crossrefs

Cf. A208913, 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] + 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[%]    (* A208913 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208914 *)

Formula

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

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

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 3, 9, 10, 5, 5, 18, 28, 22, 8, 8, 35, 68, 74, 45, 13, 13, 66, 154, 210, 177, 88, 21, 21, 122, 331, 541, 574, 397, 167, 34, 34, 222, 686, 1302, 1656, 1446, 850, 310, 55, 55, 399, 1382, 2982, 4404, 4614, 3434, 1758, 566, 89, 89, 710, 2723
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,...
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
  1;
  1,  2;
  2,  4,  3;
  3,  9, 10,  5;
  5, 18, 28, 22,  8;
First three polynomials v(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.
From _Philippe Deléham_, Apr 11 2012: (Start)
Triangle in A185081 begins:
  1;
  0,  1;
  0,  1,  2;
  0,  2,  4,  3;
  0,  3,  9, 10,  5;
  0,  5, 18, 28, 22,  8;
  ... (End)

Crossrefs

Cf. A209137, 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)
T(n,k) = A185081(n,k+1).
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(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k >= n. (End)

A209144 Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section.

Original entry on oeis.org

1, 3, 6, 1, 12, 5, 24, 16, 1, 48, 44, 7, 96, 112, 30, 1, 192, 272, 104, 9, 384, 640, 320, 48, 1, 768, 1472, 912, 200, 11, 1536, 3328, 2464, 720, 70, 1, 3072, 7424, 6400, 2352, 340, 13, 6144, 16384, 16128, 7168, 1400, 96, 1, 12288, 35840, 39680, 20736
Offset: 1

Views

Author

Clark Kimberling, Mar 06 2012

Keywords

Comments

Alternating row sums: 1,3,5,7,9,11,13,15,17,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (3,-1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 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;
   3;
   6,  1;
  12,  5;
  24, 16, 1;
First three polynomials v(n,x): 1, 3, 6 + x.
(3,-1, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, ...) begins:
    1;
    3,   0;
    6,   1,   0;
   12,   5,   0, 0;
   24,  16,   1, 0, 0;
   48,  44,   7, 0, 0, 0;
   96, 112,  30, 1, 0, 0, 0;
  192, 272, 104, 9, 0, 0, 0, 0;

Crossrefs

Cf. A209143, 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] + 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[%]    (* A209143 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209144 *)

Formula

u(n,x) = u(n-1,x) + (x+1)*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.
From Philippe Deléham, Mar 07 2012: (Start)
As triangle T(n,k) with 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+x)/(1-2*x-y*x^2).
Sum_{k=0..n} T(n,k)*x^k = A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. (End)

A209149 Triangle of coefficients of polynomials v(n,x) jointly generated with A209146; see the Formula section.

Original entry on oeis.org

1, 3, 1, 6, 5, 1, 12, 16, 7, 1, 24, 44, 30, 9, 1, 48, 112, 104, 48, 11, 1, 96, 272, 320, 200, 70, 13, 1, 192, 640, 912, 720, 340, 96, 15, 1, 384, 1472, 2464, 2352, 1400, 532, 126, 17, 1, 768, 3328, 6400, 7168, 5152, 2464, 784, 160, 19, 1, 1536, 7424
Offset: 1

Views

Author

Clark Kimberling, Mar 07 2012

Keywords

Comments

Alternating row sums: 1,2,2,2,2,2,2,2,2,2,2,2,2,...
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012
A skew triangle of A209144. - Philippe Deléham, Mar 08 2012
Riordan array ( (1 + x)/(1 - 2*x), x/(1 - 2*x) ). Cf. A118800. Matrix inverse is a signed version of A112626. - Peter Bala, Jul 17 2013

Examples

			First five rows:
   1;
   3,  1;
   6,  5,  1;
  12, 16,  7, 1;
  24, 44, 30, 9, 1;
First three polynomials v(n,x): 1, 3 + x, 6 + 5x + x^2.
v(1,x) = 1
v(2,x) = 3 + x
v(3,x) = (3 + x)*(2 + x)
v(4,x) = (3 + x)*(2 + x)^2
v(5,x) = (3 + x)*(2 + x)^3
v(n,x) = (3 + x)*(2 + x)^(n-2)for n > 1. - _Philippe Deléham_, Mar 08 2012

Crossrefs

Cf. A209144, A209146, A209148, A208510.
Cf. A084938, A112626.

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] + (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[%]    (* A209148 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209149 *)

Formula

u(n,x) = u(n-1,x) + (x+1)*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.
As DELTA-triangle:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 08 2012
As DELTA-triangle: G.f. is (1+x)/(1-2*x-yx). - Philippe Deléham, Mar 08 2012

A210039 Array of coefficients of polynomials u(n,x) jointly generated with A210040; see the Formula section.

Original entry on oeis.org

1, 3, 6, 1, 10, 5, 15, 15, 1, 21, 35, 7, 28, 70, 28, 1, 36, 126, 84, 9, 45, 210, 210, 45, 1, 55, 330, 462, 165, 11, 66, 495, 924, 495, 66, 1, 78, 715, 1716, 1287, 286, 13, 91, 1001, 3003, 3003, 1001, 91, 1, 105, 1365, 5005, 6435, 3003, 455, 15, 120, 1820
Offset: 1

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

Every term is a binomial coefficient.
Row sums: A000225
For a discussion and guide to related arrays, see A208510.

Examples

			First eight rows:
1
3
6....1
10...5
15...15....1
21...35....7
28...70....28...1
36...126...84...9
First five polynomials u(n,x):
1
3
6 + x
10 + 5x
21 + 35x + 7x^2.

Crossrefs

Cf. A034839, A210040, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
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[%]    (* A210039 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210040 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)

Formula

u(n,x)=u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also, writing the general term as T(n,m),
T(n,k)=C(n,2k) for 1<=k<=floor[(n+1)/2], for n>=1.

A207615 Triangle of coefficients of polynomials v(n,x) jointly generated with A207614; see the Formula section.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 12, 11, 5, 1, 24, 28, 17, 6, 1, 48, 68, 51, 24, 7, 1, 96, 160, 142, 82, 32, 8, 1, 192, 368, 376, 255, 122, 41, 9, 1, 384, 832, 960, 744, 417, 172, 51, 10, 1, 768, 1856, 2384, 2072, 1323, 639, 233, 62, 11, 1, 1536, 4096, 5792, 5568, 3974
Offset: 1

Views

Author

Clark Kimberling, Feb 20 2012

Keywords

Comments

Riordan array ((1 + z)/(1 - 2*z), z*(1 - z)/(1 - 2*z)). - Peter Bala, Dec 31 2015

Examples

			First five rows:
   1
   3  1
   6  4  1
  12 11  5  1
  24 28 17  6  1

Crossrefs

Cf. A207614, A208510, A003945, A011782.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207614 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207615 *)

Formula

u(n,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.

A207616 Triangle of coefficients of polynomials u(n,x) jointly generated with A207617; see the Formula section.

Original entry on oeis.org

1, 2, 4, 1, 7, 4, 11, 11, 1, 16, 25, 6, 22, 50, 22, 1, 29, 91, 63, 8, 37, 154, 154, 37, 1, 46, 246, 336, 129, 10, 56, 375, 672, 375, 56, 1, 67, 550, 1254, 957, 231, 12, 79, 781, 2211, 2211, 781, 79, 1, 92, 1079, 3718, 4719, 2288, 377, 14, 106, 1456, 6006
Offset: 1

Views

Author

Clark Kimberling, Feb 20 2012

Keywords

Comments

With offset 0, equals the stretched Riordan array ((1 - z + z^2)/(1 - z)^3, z^2/(1 - z)^2) in the notation of Corsani et al., Section 2. - Peter Bala, Dec 31 2015

Examples

			First five rows:
   1
   2
   4  1
   7  4
  11 11  1

Links

Crossrefs

Cf. A207617, A208510, A000124 (column 1).

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207616 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207617 *)

Formula

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

A207629 Triangle of coefficients of polynomials u(n,x) jointly generated with A207630; see the Formula section.

Original entry on oeis.org

1, 2, 5, 1, 11, 4, 23, 13, 1, 47, 37, 6, 95, 97, 25, 1, 191, 241, 87, 8, 383, 577, 271, 41, 1, 767, 1345, 783, 169, 10, 1535, 3073, 2143, 609, 61, 1, 3071, 6913, 5631, 2001, 291, 12, 6143, 15361, 14335, 6145, 1191, 85, 1, 12287, 33793, 35583, 17921
Offset: 1

Views

Author

Clark Kimberling, Feb 23 2012

Keywords

Comments

With offset 0, equals the stretched Riordan array ((1 - z + z^2)/(1 - 3*z + 2*z^2), z^2/(1 - 2*z)) in the notation of Corsani et al., Section 2. - Peter Bala, Dec 31 2015

Examples

			First five rows:
   1
   2
   5  1
  11  4
  23 13  1

Links

Crossrefs

Cf. A207630, A208510, A083329 (column 1).

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207629 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207630 *)

Formula

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

A208747 Triangle of coefficients of polynomials u(n,x) jointly generated with A208748; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 2, 8, 1, 2, 12, 24, 1, 2, 16, 40, 80, 1, 2, 20, 56, 160, 256, 1, 2, 24, 72, 256, 576, 832, 1, 2, 28, 88, 368, 992, 2112, 2688, 1, 2, 32, 104, 496, 1504, 3968, 7552, 8704, 1, 2, 36, 120, 640, 2112, 6464, 15232, 26880, 28160, 1, 2, 40, 136, 800
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 T(n,k) given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

Examples

			First five rows:
1
1...2
1...2...8
1...2...12...24
1...2...16...40...80
First five polynomials u(n,x):
1
1 + 2x
1 + 2x + 8x^2
1 + 2x + 12x^2 + 24x^3
1 + 2x + 16x^2 + 40x^3 + 80x^4
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 0, 0, -2, 0, 0, ...) begins :
1
1, 0
1, 2, 0
1, 2, 8, 0
1, 2, 12, 24, 0
1, 2, 16, 40, 80, 0
1, 2, 20, 56, 160, 256, 0
1, 2, 24, 72, 256, 576, 832, 0. - _Philippe Deléham_, Mar 14 2012

Crossrefs

Cf. A208748, 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] + 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[%]    (* A208747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208748 *)

Formula

u(n,x)=u(n-1,x)+2x*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) = A208342(n,k)*2^k. - Philippe Deléham, Mar 05 2012
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 14 2012
G.f.: -x*y/(-1+2*x*y-2*x^2*y+4*x^2*y^2+x). - R. J. Mathar, Aug 11 2015

A208748 Triangle of coefficients of polynomials v(n,x) jointly generated with A208747; see the Formula section.

Original entry on oeis.org

1, 0, 4, 0, 2, 12, 0, 2, 8, 40, 0, 2, 8, 40, 128, 0, 2, 8, 48, 160, 416, 0, 2, 8, 56, 208, 640, 1344, 0, 2, 8, 64, 256, 928, 2432, 4352, 0, 2, 8, 72, 304, 1248, 3840, 9088, 14080, 0, 2, 8, 80, 352, 1600, 5504, 15616, 33280, 45568, 0, 2, 8, 88, 400, 1984, 7424
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, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

Examples

			First five rows:
1
0...4
0...2...12
0...2...8...40
0...2...8...40...128
First five polynomials v(n,x):
1
4x
2x + 12x^2
2x + 8x^2 + 40x^3
2x + 8x^2 + 40x^3 + 128x^4

Crossrefs

Cf. A208747, 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] + 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[%]    (* A208747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208748 *)

Formula

u(n,x)=u(n-1,x)+2x*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) = 2^k*A208343(n,k). - Philippe Deléham, Mar 05 2012
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) = 1, T(2,0) = T(3,0) = 0, T(2,1) = 4, T(3,1) = 2, T(3,2) = 12, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 14 2012
G.f.: (-1+x-2*x*y)*x*y/(-1+x+2*x*y-2*x^2*y+4*x^2*y^2). - R. J. Mathar, Aug 11 2015
Previous Showing 11-20 of 345 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.