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 21-26 of 26 results.

A193971 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=(x+1)^n.

Original entry on oeis.org

2, 3, 5, 4, 11, 9, 5, 19, 26, 14, 6, 29, 55, 50, 20, 7, 41, 99, 125, 85, 27, 8, 55, 161, 259, 245, 133, 35, 9, 71, 244, 476, 574, 434, 196, 44, 10, 89, 351, 804, 1176, 1134, 714, 276, 54, 11, 109, 485, 1275, 2190, 2562, 2058, 1110, 375, 65, 12, 131, 649, 1925
Offset: 0

Views

Author

Clark Kimberling, Aug 10 2011

Keywords

Comments

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
2
3...5
4...11....9
5...19...26...14
6...29...55...50...20
7...41...99...125..85...27

Crossrefs

Cf. A193842, A193972.

Programs

  • Maple
    # The function 'fission' is defined in A193842.
p := (n,x) -> `if`(n=0,1,x*p(n-1,x)+n+1);
q := (n,x) -> (x+1)^n;
A193971_row := n -> fission(p, q, n);
for n from 0 to 5 do A193971_row(n) od; # Peter Luschny, Jul 23 2014
  • Mathematica
    z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
q[n_, x_] := (x + 1)^n
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193971 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193972 *)
  • Sage
    # uses[fission from A193842]
p = lambda n,x: x*p(n-1,x)+n+1 if n > 0 else 1
q = lambda n,x: (x+1)^n
A193971_row = lambda n: fission(p, q, n);
for n in range(7): A193971_row(n) # Peter Luschny, Jul 23 2014

A193975 Triangular array: the self-fission of (p(n,x)), where p(n,x)=x*p(n-1,x)+n+1, where p(0,x)=1.

Original entry on oeis.org

2, 3, 8, 4, 11, 20, 5, 14, 26, 40, 6, 17, 32, 50, 70, 7, 20, 38, 60, 85, 112, 8, 23, 44, 70, 100, 133, 168, 9, 26, 50, 80, 115, 154, 196, 240, 10, 29, 56, 90, 130, 175, 224, 276, 330, 11, 32, 62, 100, 145, 196, 252, 312, 375, 440, 12, 35, 68, 110, 160, 217, 280
Offset: 0

Views

Author

Clark Kimberling, Aug 10 2011

Keywords

Comments

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
2
3...8
4...11...20
5...14...26...40
6...17...32...50...70
7...20...38...60...85...112

Crossrefs

Cf. A193842, A193976.

Programs

  • Mathematica
    z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193975 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193976 *)

A193979 Triangular array: the self-fission of (p(n,x)), where p(n,x)=x*p(n-1,x)+n, with p(0,x)=1.

Original entry on oeis.org

1, 2, 3, 3, 5, 9, 4, 7, 13, 21, 5, 9, 17, 28, 41, 6, 11, 21, 35, 52, 71, 7, 13, 25, 42, 63, 87, 113, 8, 15, 29, 49, 74, 103, 135, 169, 9, 17, 33, 56, 85, 119, 157, 198, 241, 10, 19, 37, 63, 96, 135, 179, 227, 278, 331, 11, 21, 41, 70, 107, 151, 201, 256, 315, 377
Offset: 0

Views

Author

Clark Kimberling, Aug 10 2011

Keywords

Comments

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
1
2...3
3...5...9
4...7...13....21
5...9...17....28...41
6...11...21...35...52...71

Crossrefs

Cf. A193842, A193980.

Programs

  • Mathematica
    z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n;
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193979 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193980 *)

A193997 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers) and q(n,x)=(x+1)^n.

Original entry on oeis.org

1, 2, 3, 3, 8, 6, 5, 18, 23, 11, 8, 37, 66, 55, 19, 13, 73, 167, 196, 120, 32, 21, 139, 388, 587, 511, 246, 53, 34, 259, 853, 1578, 1777, 1225, 484, 87, 55, 474, 1799, 3933, 5428, 4857, 2765, 924, 142, 89, 856, 3678, 9275, 15147, 16642, 12333, 5969
Offset: 0

Views

Author

Clark Kimberling, Aug 11 2011

Keywords

Comments

See A193842 for the definition of the fission of P by Q, where P and Q are sequences of polynomials or triangular arrays (of coefficients of polynomials).

Examples

			First six rows:
1
2....3
3....8....6
5....18...23....11
8....37...66....55....19
13...73...167...196...120...32

Crossrefs

Cf. A193842, A193998.

Programs

  • Mathematica
    z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := (x + 1)^n;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193997 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193998 *)

A194007 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers) and q(n,x)=x*q(n-1,x)+n+1, with q(0,x)=1.

Original entry on oeis.org

1, 2, 5, 3, 8, 14, 5, 13, 23, 34, 8, 21, 37, 55, 74, 13, 34, 60, 89, 120, 152, 21, 55, 97, 144, 194, 246, 299, 34, 89, 157, 233, 314, 398, 484, 571, 55, 144, 254, 377, 508, 644, 783, 924, 1066, 89, 233, 411, 610, 822, 1042, 1267, 1495, 1725, 1956, 144, 377
Offset: 0

Views

Author

Clark Kimberling, Aug 11 2011

Keywords

Comments

See A193842 for the definition of the fission of P by Q, where P and Q are sequences of polynomials or triangular arrays (of coefficients of polynomials).

Examples

			First six rows:
1
2....5
3....8....14
5....13...23...34
8....21...37...55...74
13...34...60...89...120...152

Crossrefs

Cf. A193842, A194008.

Programs

  • Mathematica
    z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := x*q[n - 1, x] + n + 1; q[0, n_] := 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194007 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194008 *)

A194009 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

2, 3, 5, 4, 7, 13, 5, 9, 17, 28, 6, 11, 21, 35, 58, 7, 13, 25, 42, 70, 114, 8, 15, 29, 49, 82, 134, 218, 9, 17, 33, 56, 94, 154, 251, 407, 10, 19, 37, 63, 106, 174, 284, 461, 747, 11, 21, 41, 70, 118, 194, 317, 515, 835, 1352, 12, 23, 45, 77, 130, 214, 350, 569
Offset: 0

Views

Author

Clark Kimberling, Aug 11 2011

Keywords

Comments

See A193842 for the definition of the fission of P by Q, where P and Q are sequences of polynomials or triangular arrays (of coefficients of polynomials).

Examples

			First six rows:
2
3...5
4...7....13
5...9....17...28
6...11...21...35...58
7...13...25...42...70...114

Crossrefs

Cf. A193842, A194010.

Programs

  • Mathematica
    z = 11;
p[n_, x_] := x*p[n - 1, x] + n + 1; p[0, n_] := 1;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194009 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194010 *)
Previous Showing 21-26 of 26 results.

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