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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A193727 Mirror of the triangle A193726.

Original entry on oeis.org

1, 2, 1, 10, 9, 2, 50, 65, 28, 4, 250, 425, 270, 76, 8, 1250, 2625, 2200, 920, 192, 16, 6250, 15625, 16250, 9000, 2800, 464, 32, 31250, 90625, 112500, 77500, 32000, 7920, 1088, 64, 156250, 515625, 743750, 612500, 315000, 103600, 21280, 2496, 128
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

This triangle is obtained by reversing the rows of the triangle A193726.
Triangle T(n,k), read by rows, given by (2,3,0,0,0,0,0,0,0,...) DELTA (1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
     1;
     2,    1;
    10,    9,    2;
    50,   65,   28,   4;
   250,  425,  270,  76,   8;
  1250, 2625, 2200, 920, 192; 16;

Links

Crossrefs

Cf. A011782, A015535, A020699, A081040, A084938.
Cf. A107839, A133494, A169634, A193722, A193726.

Programs

  • Magma
    function T(n, k) // T = A193727
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 5*T(n-1, k) + 2*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2023
  • Mathematica
    (* First program *)
z = 8; a = 1; b = 2; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193726 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 5*T[n-1, k] + 2*T[n-1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2023 *)
  • SageMath
    def T(n, k): # T = A193727
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 5*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 02 2023

Formula

T(n,k) = A193726(n,n-k).
T(n,k) = 2*T(n-1,k-1) + 5*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-3*x-x*y)/(1-5*x-2*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 02 2023: (Start)
T(n, 0) = A020699(n).
T(n, 1) = A081040(n-1).
T(n, n) = A011782(n).
Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].
Sum_{k=0..n} (-1)^k * T(n, k) = A133494(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = 2*A015535(n) + A015535(n-1) + (1/2)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = 2*A107839(n-1) - A107839(n-2) + (1/2)*[n=0]. (End)

A193728 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+1)^n.

Original entry on oeis.org

1, 2, 1, 8, 10, 3, 32, 64, 42, 9, 128, 352, 360, 162, 27, 512, 1792, 2496, 1728, 594, 81, 2048, 8704, 15360, 14400, 7560, 2106, 243, 8192, 40960, 87552, 103680, 73440, 31104, 7290, 729, 32768, 188416, 473088, 677376, 604800, 344736, 122472, 24786, 2187
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
Triangle T(n,k), read by rows, given by (2,2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
    1;
    2,    1;
    8,   10,    3;
   32,   64,   42,    9;
  128,  352,  360,  162,  27;
  512, 1792, 2496, 1728, 594, 81;

Links

Crossrefs

Cf. A000012, A000420, A060816, A081038, A081294.
Cf. A084938, A133494, A193722, A193729.

Programs

  • Magma
    function T(n, k) // T = A193728
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 4*T(n-1, k) + 3*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 28 2023
  • Mathematica
    (* First program *)
z = 8; a = 1; b = 2; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193728 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193729 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*T[n-1,k] + 3*T[n-1,k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 28 2023 *)
  • SageMath
    def T(n, k): # T = A193728
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 4*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 28 2023

Formula

T(n,k) = 3*T(n-1,k-1) + 4*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x-2*x*y)/(1-4*x-3*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 28 2023: (Start)
T(n, n-k) = A193729(n, k).
T(n, 0) = A081294(n).
T(n, n-1) = 2*A081038(n-1).
T(n, n) = A133494(n).
Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).
Sum_{k=0..n} (-1)^k * T(n, k) = A000012(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (5*b(n) + 4*b(n-1))/14 + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A060816(n),
where b(n) = (2 + sqrt(7))^n + (2 - sqrt(7))^n. (End)

A193729 Mirror of the triangle A193728.

Original entry on oeis.org

1, 1, 2, 3, 10, 8, 9, 42, 64, 32, 27, 162, 360, 352, 128, 81, 594, 1728, 2496, 1792, 512, 243, 2106, 7560, 14400, 15360, 8704, 2048, 729, 7290, 31104, 73440, 103680, 87552, 40960, 8192, 2187, 24786, 122472, 344736, 604800, 677376, 473088, 188416, 32768
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

T(n, k) is obtained by reversing the rows of the triangle A193728.
Triangle T(n,k), read by rows, given by [1,2,0,0,0,0,...] DELTA [2,2,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
   1;
   1,   2;
   3,  10,    8;
   9,  42,   64,   32;
  27, 162,  360,  352,  128;
  81, 594, 1728, 2496, 1792, 512;

Links

Crossrefs

Cf. A000420, A033999, A081038, A081294, A084938.
Cf. A108981, A133494, A193722, A193728, A247560.

Programs

  • Magma
    function T(n, k) // T = A193729
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 3*T(n-1, k) + 4*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 28 2023
  • Mathematica
    (* First program *)
z = 8; a = 1; b = 2; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193728 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193729 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 3*T[n-1,k] + 4*T[n -1, k-1]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 28 2023 *)
  • SageMath
    def T(n, k): # T = A193729
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 3*T(n-1, k) + 4*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 28 2023

Formula

Let w(n,k) be the triangle of A193728, then the triangle in this sequence is given by w(n,n-k).
T(n,k) = 4*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x-2*x*y)/(1-3*x-4*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 28 2023: (Start)
T(n, 0) = A133494(n).
T(n, 1) = 2*A081038(n-1).
T(n, n) = A081294(n).
Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).
Sum_{k=0..n} (-1)^k * T(n, k) = A033999(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*[n=0] + A108981(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = (1/2)*[n=0] + A247560(n-1).
(End)

A193730 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.

Original entry on oeis.org

1, 2, 1, 4, 8, 3, 8, 28, 30, 9, 16, 80, 144, 108, 27, 32, 208, 528, 648, 378, 81, 64, 512, 1680, 2880, 2700, 1296, 243, 128, 1216, 4896, 10800, 14040, 10692, 4374, 729, 256, 2816, 13440, 36288, 60480, 63504, 40824, 14580, 2187, 512, 6400, 35328, 112896, 229824, 308448, 272160, 151632, 48114, 6561
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
Triangle T(n,k), read by rows, given by (2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
   1;
   2,   1;
   4,   8,   3;
   8,  28,  30,   9;
  16,  80, 144, 108,  27;
  32, 208, 528, 648, 378, 81;

Links

Crossrefs

Cf. A000079, A005053, A084938, A130129, A133494, A165326.
Cf. A193722, A193731, A199923.

Programs

  • Magma
    function T(n, k) // T = A193730
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 2*T(n-1, k) + 3*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023
  • Mathematica
    (* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]     (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 2*T[n-1, k] + 3*T[n-1, k-1]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)
  • SageMath
    def T(n, k): # T = A193730
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 2*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023

Formula

T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x*y)/(1-2*x-3*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 20 2023: (Start)
T(n, 0) = A000079(n).
T(n, 1) = A130129(n-1).
T(n, n) = A133494(n).
T(n, n-1) = A199923(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A165326(n). (End)

A193731 Mirror of the triangle A193730.

Original entry on oeis.org

1, 1, 2, 3, 8, 4, 9, 30, 28, 8, 27, 108, 144, 80, 16, 81, 378, 648, 528, 208, 32, 243, 1296, 2700, 2880, 1680, 512, 64, 729, 4374, 10692, 14040, 10800, 4896, 1216, 128, 2187, 14580, 40824, 63504, 60480, 36288, 13440, 2816, 256, 6561, 48114, 151632, 272160, 308448, 229824, 112896, 35328, 6400, 512
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

A193731 is obtained by reversing the rows of the triangle A193730.
Triangle T(n,k), read by rows, given by (1,2,0,0,0,0,0,0,0,...) DELTA (2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
   1;
   1,   2;
   3,   8,   4;
   9,  30,  28,   8;
  27, 108, 144,  80,  16;
  81, 378, 648, 528, 208, 32;

Links

Crossrefs

Cf. A000012, A000079, A005053, A006234, A007483, A080420, A080421.
Cf. A080422, A080423, A084938, A130129, A133494, A153881.
Cf. A193722, A193730.

Programs

  • Magma
    function T(n, k) // T = A193731
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 3*T(n-1, k) + 2*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023
  • Mathematica
    (* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 3*T[n-1, k] + 2*T[n -1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)
  • SageMath
    def T(n, k): # T = A193731
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 3*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023

Formula

T(n,k) = A193730(n,n-k).
T(n,k) = 2*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x)/(1-3*x-2*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 20 2023: (Start)
T(n, 0) = A133494(n).
T(n, 1) = 2*A006234(n+2).
T(n, 2) = 4*A080420(n-2).
T(n, 3) = 8*A080421(n-3).
T(n, 4) = 16*A080422(n-4).
T(n, 5) = 32*A080423(n-5).
T(n, n) = A000079(n).
T(n, n-1) = A130129(n-1).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A153881(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007483(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000012(n). (End)

A193735 Mirror of the triangle A193734.

Original entry on oeis.org

1, 2, 1, 8, 6, 1, 32, 32, 10, 1, 128, 160, 72, 14, 1, 512, 768, 448, 128, 18, 1, 2048, 3584, 2560, 960, 200, 22, 1, 8192, 16384, 13824, 6400, 1760, 288, 26, 1, 32768, 73728, 71680, 39424, 13440, 2912, 392, 30, 1, 131072, 327680, 360448, 229376, 93184, 25088, 4480, 512, 34, 1
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

A193735 is obtained by reversing the rows of the triangle A193734.
Triangle T(n,k), read by rows, given by (2,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
    1;
    2,   1;
    8,   6,   1;
   32,  32,  10,   1;
  128, 160,  72,  14,  1;
  512, 768, 448, 128, 18, 1;

Links

Crossrefs

Cf. A084938, A193722, A193734.
Cf. A001075, A001077, A005053, A081294, A133494.

Programs

  • Magma
    function T(n, k) // T = A193735
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 4*T(n-1, k) + T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2023
  • Mathematica
    (* First program *)
z = 8; a = 2; b = 1; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193734 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]      (* A193735 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*T[n-1, k] + T[n -1, k-1]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2023 *)
  • SageMath
    def T(n, k): # T = A193735
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 4*T(n-1, k) + T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 19 2023

Formula

T(n,k) = A193734(n,n-k).
T(n,k) = T(n-1,k-1) + 4*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x)/(1-4*x-x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 19 2023: (Start)
T(n, 0) = A081294(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A133494(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001077(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A001075(n). (End)

A193736 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (n+1)-st Fibonacci polynomial and q(n,x) = (x+1)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 8, 8, 3, 1, 5, 13, 19, 15, 5, 1, 6, 19, 36, 42, 28, 8, 1, 7, 26, 60, 91, 89, 51, 13, 1, 8, 34, 92, 170, 216, 182, 92, 21, 1, 9, 43, 133, 288, 446, 489, 363, 164, 34, 1, 10, 53, 184, 455, 826, 1105, 1068, 709, 290, 55, 1, 11, 64, 246, 682, 1414, 2219, 2619, 2266, 1362, 509, 89
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  4,  2;
  1,  4,  8,  8,  3;
  1,  5, 13, 19, 15,  5;

Links

Crossrefs

Cf. A000007, A005314 (diagonal sums), A052542 (row sums), A077962.
Cf. A029907, A193722, A193737, A324969.

Programs

  • Magma
    function T(n,k) // T = A193736
  if k lt 0 or n lt 0 then return 0;
  elif n lt 3 then return Binomial(n,k);
  else return T(n-1, k) + T(n-1, k-1) + T(n-2, k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023
  • Mathematica
    (* First program *)
p[0, x_] := 1
p[n_, x_] := Fibonacci[n + 1, x] /; n > 0
q[n_, x_] := (x + 1)^n
t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
t[n_, n_] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n - k + 1, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193736 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]          (* A193737 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<3, Binomial[n, k], T[n-1,k] +T[n-1,k-1] +T[n -2,k-2]];
Table[T[n, k], {n,0,12}, {k,0,n}]//TableForm (* G. C. Greubel, Oct 24 2023 *)
  • SageMath
    def T(n,k): # T = A193736
    if (n<3): return binomial(n,k)
    else: return T(n-1,k) +T(n-1,k-1) +T(n-2,k-2)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

Formula

T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,2) = 1; T(n,k) = 0 if k<0 or k>n; T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2). - Philippe Deléham, Feb 13 2020
From G. C. Greubel, Oct 24 2023: (Start)
T(n, k) = A193737(n, n-k).
T(n, n) = Fibonacci(n) + [n=0] = A324969(n+1).
T(n, n-1) = A029907(n).
Sum_{k=0..n} T(n, k) = A052542(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A005314(n) + [n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = [n=0] + A077962(n-1). (End)

A193740 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=q(n,x) in A193738.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 4, 9, 9, 1, 4, 10, 19, 19, 1, 4, 10, 20, 34, 34, 1, 4, 10, 20, 35, 55, 55, 1, 4, 10, 20, 35, 56, 83, 83, 1, 4, 10, 20, 35, 56, 84, 119, 119, 1, 4, 10, 20, 35, 56, 84, 120, 164, 164, 1, 4, 10, 20, 35, 56, 84, 120, 165, 219, 219, 1, 4, 10, 20, 35, 56
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
1
1....1
1....3....3
1....4....9....9
1....4....10....19...19
1....4....10....20...34...34

Crossrefs

Cf. A193722, A193741.

Programs

  • Mathematica
    z = 12;
p[0, x_] := 1
p[n_, x_] := n + Sum[(k + 1) x^(n - k), {k, 0, n - 1}]
q[n_, x_] := p[n, x]
t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
t[n_, n_] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193740 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193741 *)

A193790 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=1+x^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 4, 9, 1, 6, 12, 8, 27, 1, 8, 24, 32, 16, 81, 1, 10, 40, 80, 80, 32, 243, 1, 12, 60, 160, 240, 192, 64, 729, 1, 14, 84, 280, 560, 672, 448, 128, 2187, 1, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 6561, 1, 18, 144, 672, 2016, 4032, 5376
Offset: 0

Views

Author

Clark Kimberling, Aug 05 2011

Keywords

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
1
1....1
1....2....3
1....4....4....9
1....6....12....8...27
1....8....24....32...16...81

Crossrefs

Cf. A193791.

Programs

  • Mathematica
    z = 10; a = 2; b = 1;
p[n_, x_] := (a*x + b)^n
q[n_, x_] := 1 + x^n ; q[n_, 0] := q[n, x] /. x -> 0;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193790 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193791 *)

A193818 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = x^n + x^(n-1) + ... + x+1 and q(n,x)=(2x+1)^n.

Original entry on oeis.org

1, 2, 1, 4, 6, 2, 8, 16, 12, 3, 16, 40, 40, 20, 4, 32, 96, 120, 80, 30, 5, 64, 224, 336, 280, 140, 42, 6, 128, 512, 896, 896, 560, 224, 56, 7, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 8, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 9, 1024, 5632
Offset: 0

Views

Author

Clark Kimberling, Aug 06 2011

Keywords

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
Triangle T(n,k), read by rows, given by (2,0,-2,2,0,0,0,0,0,0,0,...) DELTA (1,1,-1,1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
   1;
   2,  1;
   4,  6,   2;
   8, 16,  12,  3;
  16, 40,  40, 20,  4;
  32, 96, 120, 80, 30, 5;

Crossrefs

Cf. A084938, A193815, A193722, A193819.

Programs

  • Mathematica
    z = 10; c = 2; d = 1;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193818 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193819 *)

Formula

T(n,k) = A193815(n,k)*2(n-k).
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) - T(n-2,k-2), T(0,0)=T(1,1)=1, T(1,0)=2, T(2,0)=4, T(2,1)=6, T(2,2)=2, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Dec 15 2013
G.f.: (1-x*y+2*x^2*y+x^2*y^2)/((-1+2*x+x*y)*(x*y-1)). - R. J. Mathar, Aug 11 2015
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