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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A209999 Triangle of coefficients of polynomials u(n,x) jointly generated with A210287; see the Formula section.

Original entry on oeis.org

1, 2, 2, 4, 6, 3, 7, 16, 13, 4, 12, 36, 44, 24, 5, 20, 76, 122, 100, 40, 6, 33, 152, 306, 332, 201, 62, 7, 54, 294, 712, 968, 783, 370, 91, 8, 88, 554, 1573, 2572, 2614, 1666, 637, 128, 9, 143, 1024, 3339, 6392, 7829, 6296, 3277, 1040, 174, 10, 232, 1864
Offset: 1

Views

Author

Clark Kimberling, Mar 23 2012

Keywords

Comments

Column 1: -1+F(n+2), where F=000045 (Fibonacci numbers)
Row sums: A003462
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
4....6....3
7....16...13...4
12...36...44...24...5
First three polynomials u(n,x): 1, 2 + 2x, 4 + 6x + 3x^2.

Crossrefs

Cf. A210287, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%]    (* A209999 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210287 *)

Formula

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

A210038 Triangle of coefficients of polynomials v(n,x) jointly generated with A210037; see the Formula section.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 15, 12, 5, 1, 31, 32, 18, 6, 1, 63, 80, 56, 25, 7, 1, 127, 192, 160, 88, 33, 8, 1, 255, 448, 432, 280, 129, 42, 9, 1, 511, 1024, 1120, 832, 450, 180, 52, 10, 1, 1023, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 2047, 5120, 6912, 6400
Offset: 1

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

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

Examples

			First five rows:
1
3....1
7....4....1
15...12...5....1
31...32...18...6...1
First three polynomials v(n,x): 1, 3 + x , 7 + 4x + x^2.

Crossrefs

Cf. A210037, 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_] := 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[%]    (* A210037 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210038 *)

Formula

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

A210197 Triangle of coefficients of polynomials u(n,x) jointly generated with A210198; see the Formula section.

Original entry on oeis.org

1, 3, 7, 1, 15, 5, 31, 17, 1, 63, 49, 7, 127, 129, 31, 1, 255, 321, 111, 9, 511, 769, 351, 49, 1, 1023, 1793, 1023, 209, 11, 2047, 4097, 2815, 769, 71, 1, 4095, 9217, 7423, 2561, 351, 13, 8191, 20481, 18943, 7937, 1471, 97, 1, 16383, 45057, 47103
Offset: 1

Views

Author

Clark Kimberling, Mar 18 2012

Keywords

Comments

Column 1: -1+2^n
Row sums: A048739
Alternating row sums: triangular numbers, A000217
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
3
7....1
15...5
31...17...1
First three polynomials u(n,x): 1, 3, 7 + x.

Links

Crossrefs

Cf. A210198, A208510.
Essentially the same as the triangle in A257597.

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 + 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[%]   (* A210197 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210198 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A048739 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A005409 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000217 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)

Formula

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

A210204 Triangle of coefficients of polynomials v(n,x) jointly generated with A210203; see the Formula section.

Original entry on oeis.org

1, 3, 2, 7, 8, 2, 15, 24, 12, 2, 31, 64, 48, 16, 2, 63, 160, 160, 80, 20, 2, 127, 384, 480, 320, 120, 24, 2, 255, 896, 1344, 1120, 560, 168, 28, 2, 511, 2048, 3584, 3584, 2240, 896, 224, 32, 2, 1023, 4608, 9216, 10752, 8064, 4032, 1344, 288, 36, 2, 2047
Offset: 1

Views

Author

Clark Kimberling, Mar 18 2012

Keywords

Comments

Column 1: -1+2^n.
Row sums: A048473.
Alternating row sums: 1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Row sums without first column give A056182. - Alois P. Heinz, Jan 14 2022

Examples

			First five rows:
1
3....2
7....8....2
15...24...12...2
31...64...48...16...2
First three polynomials v(n,x): 1, 3 + 2x , 7 + 8x + 2x^2.

Crossrefs

Cf. A210203, A208510.
Cf. A056182.

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 + 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[%]   (* A210203 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210204 *)

Formula

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

A210217 Triangle of coefficients of polynomials u(n,x) jointly generated with A210218; see the Formula section.

Original entry on oeis.org

1, 2, 1, 2, 5, 1, 2, 6, 12, 1, 2, 6, 19, 27, 1, 2, 6, 20, 57, 58, 1, 2, 6, 20, 67, 160, 121, 1, 2, 6, 20, 68, 218, 424, 248, 1, 2, 6, 20, 68, 231, 680, 1073, 503, 1, 2, 6, 20, 68, 232, 775, 2028, 2619, 1014, 1, 2, 6, 20, 68, 232, 791, 2543, 5797, 6214, 2037, 1, 2
Offset: 1

Views

Author

Clark Kimberling, Mar 19 2012

Keywords

Comments

Limiting row: A006012
Row sums: even-indexed Fibonacci numbers: 1,3,8,21,...
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...1
2...5...1
2...6...12...1
2...6...19...27...1
First three polynomials u(n,x): 1, 2 + x, 2 + 5x + x^2.

Crossrefs

Cf. A210218, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*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[%]    (* A210217 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210218 *)

Formula

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

A210220 T(n, k) = -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2). Triangle read by rows, T(n, k) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 13, 4, 5, 20, 34, 24, 5, 6, 30, 70, 80, 40, 6, 7, 42, 125, 200, 166, 62, 7, 8, 56, 203, 420, 496, 314, 91, 8, 9, 72, 308, 784, 1211, 1106, 553, 128, 9, 10, 90, 444, 1344, 2576, 3108, 2269, 920, 174, 10, 11, 110, 615, 2160, 4956, 7476, 7274, 4352, 1461, 230, 11
Offset: 1

Views

Author

Clark Kimberling, Mar 19 2012

Keywords

Comments

Previous name: Triangle of coefficients of polynomials v(n,x) jointly generated with A210217.
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
  1
  2...2
  3...6....3
  4...12...13...4
  5...20...34...24...5
First three polynomials v(n,x): 1, 2 + 2x , 3 + 6x + 3x^2.

Crossrefs

Cf. A210217, A210219, A208510, A016061, A112742.

Programs

  • Maple
    T := (n,k) -> -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2):
seq(seq(simplify(T(n,k)), k=1..n), n=1..10); # Peter Luschny, Oct 31 2019
  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := 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[%]     (* A210219 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210220 *)
(* alternate program *)
T[n_,k_]:=Sum[Binomial[2*j+k-2,k-1],{j,1,n-k+1}];Flatten[Table[T[n,k],{n,1,11},{k,1,n}]] (* Detlef Meya, Dec 05 2023 *)

Formula

First and last term in row n: n.
Column 2: n*(n-1).
Column 3: A016061.
Column 4: A112742.
Row sums: -1+(even-indexed Fibonacci numbers).
Periodic alternating row sums: 1,0,0,1,0,0,1,0,0,...
u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = Sum_{j=1..n-k+1} binomial(2*j+k-2,k-1). - Detlef Meya, Dec 05 2023

Extensions

New name from Peter Luschny, Oct 31 2019

A210221 Triangle of coefficients of polynomials u(n,x) jointly generated with A210596; see the Formula section.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 4, 8, 10, 8, 8, 13, 20, 24, 16, 16, 21, 40, 52, 56, 32, 32, 34, 76, 116, 128, 128, 64, 64, 55, 142, 240, 312, 304, 288, 128, 128, 89, 260, 488, 688, 800, 704, 640, 256, 256, 144, 470, 964, 1496, 1856, 1984, 1600, 1408, 512, 512, 233, 840
Offset: 1

Views

Author

Clark Kimberling, Mar 24 2012

Keywords

Comments

Row sums: even-indexed Fibonacci numbers.
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, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

Examples

			First five rows:
  1;
  2;
  3,  2;
  5,  4, 4;
  8, 10, 8, 8;
First three polynomials u(n,x):
  1
  2
  3 + 2x.
From _Philippe Deléham_, Mar 25 2012: (Start)
(1, 1, -1, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, ...) begins:
   1;
   1,  0;
   2,  0,  0;
   3,  2,  0,  0;
   5,  4,  4,  0,  0;
   8, 10,  8,  8,  0,  0;
  13, 20, 24, 16, 16,  0,  0; (End)

Links

Crossrefs

Cf. A210596, A208510.

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] + 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[%]   (* A210221 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210596 *)
With[{m = 10}, Rest[CoefficientList[CoefficientList[Series[(1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2), {x, 0, m}, {y, 0, m}], x], y]]]//Flatten (* G. C. Greubel, Dec 16 2018 *)
T[n_, k_]:= If[k < 0 || k > n, 0, T[n-1, k] + 2*T[n-1, k-1] + T[n-2, k] - 2*T[n-2, k-1]]; T[1, 0] = 1 ; T[2, 0] = 2; T[2, 1] = 0; Join[{1}, Table[T[n, k], {n, 1, 10}, {k, 0, n-2}]//Flatten] (* G. C. Greubel, Dec 17 2018 *)
  • Python
    from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + 2*x*v(n - 1, x)
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

Formula

u(n,x) = u(n-1,x) + v(n-1,x),
v(n,x) = u(n-1,x) + 2*x*v(n-1,x) [Corrected by Indranil Ghosh, May 27 2017]
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 25 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 2, T(1,1) = T(2,1) = T(2,2) = 0, T(n,k) = 0 if k < 0 or if k >= n. (End)

A210225 Triangle of coefficients of polynomials u(n,x) jointly generated with A210226; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 12, 10, 1, 5, 22, 36, 17, 1, 6, 35, 88, 87, 26, 1, 7, 51, 175, 277, 181, 37, 1, 8, 70, 306, 680, 734, 338, 50, 1, 9, 92, 490, 1416, 2196, 1710, 582, 65, 1, 10, 117, 736, 2632, 5402, 6156, 3606, 941, 82, 1, 11, 145, 1053, 4502, 11592
Offset: 1

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

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

Examples

			First five rows:
1
2...1
3...5....1
4...12...10...1
5...22...36...17...1
First three polynomials u(n,x): 1, 2 + x, 3 + 5x + x^2.

Crossrefs

Cf. A210226, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := 2 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[%]     (* A210225 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210226 *)

Formula

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

A210226 Triangle of coefficients of polynomials v(n,x) jointly generated with A210225; see the Formula section.

Original entry on oeis.org

1, 2, 3, 3, 9, 5, 4, 18, 24, 7, 5, 30, 66, 51, 9, 6, 45, 140, 189, 94, 11, 7, 63, 255, 505, 457, 157, 13, 8, 84, 420, 1110, 1516, 976, 244, 15, 9, 108, 644, 2142, 3986, 3960, 1896, 359, 17, 10, 135, 936, 3766, 8960, 12338, 9276, 3419, 506, 19, 11, 165
Offset: 1

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

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

Examples

			First five rows:
1
2...3
3...9....5
4...18...24...7
5...30...66...51...9
First three polynomials v(n,x): 1, 2 + 3x , 3 + 9x + 5x^2.

Crossrefs

Cf. A210225, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := 2 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[%]     (* A210225 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210226 *)

Formula

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

A210235 Triangle of coefficients of polynomials u(n,x) jointly generated with A210236; see the Formula section.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 7, 12, 7, 1, 12, 29, 28, 11, 1, 20, 64, 86, 56, 16, 1, 33, 132, 230, 210, 101, 22, 1, 54, 261, 560, 662, 451, 169, 29, 1, 88, 500, 1279, 1860, 1646, 883, 267, 37, 1, 143, 936, 2785, 4819, 5257, 3682, 1611, 403, 46, 1, 232, 1721, 5848
Offset: 1

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

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

Examples

			First five rows:
1
2....1
4....4....1
7....12...7....1
12...29...28...11...1
First three polynomials u(n,x): 1, 2 + x, 4 + 4x + x^2.

Crossrefs

Cf. A210236, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
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[%]      (* A210235 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]      (* A210236 *)

Formula

u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
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.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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