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 11-14 of 14 results.

A173076 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(floor(n/2))*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 7, 13, 7, 1, 1, 8, 21, 21, 8, 1, 1, 13, 46, 67, 46, 13, 1, 1, 14, 60, 114, 114, 60, 14, 1, 1, 23, 123, 295, 389, 295, 123, 23, 1, 1, 24, 147, 419, 685, 685, 419, 147, 24, 1, 1, 41, 300, 1015, 2001, 2491, 2001, 1015, 300, 41, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 09 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  3,   1;
  1,  4,   4,    1;
  1,  7,  13,    7,    1;
  1,  8,  21,   21,    8,    1;
  1, 13,  46,   67,   46,   13,    1;
  1, 14,  60,  114,  114,   60,   14,    1;
  1, 23, 123,  295,  389,  295,  123,   23,   1;
  1, 24, 147,  419,  685,  685,  419,  147,  24,  1;
  1, 41, 300, 1015, 2001, 2491, 2001, 1015, 300, 41, 1;

Links

Crossrefs

Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173077 (q=3).

Programs

  • Magma
    T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^(Floor(n/2))*Binomial(n-2,k-1) -1 >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021
  • Mathematica
    T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(Floor[n/2])*Binomial[n-2, k-1]];
Table[T[n, k, 2], {n,0,10}, {k,0,n}]//Flatten
  • Sage
    def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^(n//2)*binomial(n-2,k-1) -1
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)])

Formula

T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 2.

Extensions

Edited by G. C. Greubel, Jul 09 2021

A173077 Triangle T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 12, 23, 12, 1, 1, 13, 36, 36, 13, 1, 1, 32, 122, 181, 122, 32, 1, 1, 33, 155, 304, 304, 155, 33, 1, 1, 88, 513, 1270, 1689, 1270, 513, 88, 1, 1, 89, 602, 1784, 2960, 2960, 1784, 602, 89, 1, 1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 09 2010

Keywords

Examples

			Triangle starts:
  1;
  1,   1;
  1,   4,    1;
  1,   5,    5,    1;
  1,  12,   23,   12,     1;
  1,  13,   36,   36,    13,     1;
  1,  32,  122,  181,   122,    32,     1;
  1,  33,  155,  304,   304,   155,    33,    1;
  1,  88,  513, 1270,  1689,  1270,   513,   88,    1;
  1,  89,  602, 1784,  2960,  2960,  1784,  602,   89,   1;
  1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1;
  ...
Row sums: 1, 2, 6, 12, 49, 100, 491, 986, 5433, 10872, 63223, ...

Links

Crossrefs

Cf. A132044 (q=0), A173075 (q=1), A173076 (q=2), this sequence (q=3).

Programs

  • Magma
    T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^(Floor(n/2))*Binomial(n-2,k-1) -1 >;
[T(n,k,3): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021
  • Mathematica
    T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + 3^Floor[n/2] Binomial[n-2, k- 1]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
  • Sage
    def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^(n//2)*binomial(n-2,k-1) -1
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021

Formula

T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.

A173043 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 10, 10, 1, 1, 19, 261, 19, 1, 1, 36, 32777, 32777, 36, 1, 1, 69, 16777230, 68719476755, 16777230, 69, 1, 1, 134, 34359738388, 1180591620717411303458, 1180591620717411303458, 34359738388, 134, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 08 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  5,        1;
  1, 10,       10,           1;
  1, 19,      261,          19,        1;
  1, 36,    32777,       32777,       36,  1;
  1, 69, 16777230, 68719476755, 16777230, 69, 1;

Links

Crossrefs

Cf. A132044 (q=0), A007318 (q=1), this sequence (q=2), A173045 (q=3).
Cf. A000295.

Programs

  • Magma
    T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) -1 +q^(n*Binomial(n-2, k-1)) >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2021
  • Mathematica
    T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(n*Binomial[n-2, k-1])];
Table[t[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *)
  • Sage
    def T(n,k,q):
    if (k==0 or k==n): return 1
    else: return binomial(n,k) -1 +q^(n*binomial(n-2, k-1))
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2021

Formula

T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 2.
Sum_{k=0..n} T(n, k, 2) = A000295(n) + Sum_{k=0..n} 2^(n*binomial(n-2, k-1)). - G. C. Greubel, Feb 19 2021

Extensions

Edited by G. C. Greubel, Feb 19 2021

A173045 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 6566, 84, 1, 1, 247, 14348916, 14348916, 247, 1, 1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1, 1, 2193, 50031545098999727, 2503155504993241601315571986085883, 2503155504993241601315571986085883, 50031545098999727, 2193, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 08 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,   1;
  1,  10,            1;
  1,  29,           29,                  1;
  1,  84,         6566,                 84,            1;
  1, 247,     14348916,           14348916,          247,   1;
  1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1;

Links

Crossrefs

Cf. A132044 (q=0), A007318 (q=1), A173043 (q=2), this sequence (q=3).
Cf. A000295.

Programs

  • Magma
    T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) -1 +q^(n*Binomial(n-2, k-1)) >;
[T(n,k,3): k in [0..n], n in [0..9]]; // G. C. Greubel, Feb 19 2021
  • Mathematica
    T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(n*Binomial[n-2, k-1])];
Table[t[n, k, 3], {n,0,9}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *)
  • Sage
    def T(n,k,q):
    if (k==0 or k==n): return 1
    else: return binomial(n,k) -1 +q^(n*binomial(n-2, k-1))
flatten([[T(n,k,3) for k in (0..n)] for n in (0..9)]) # G. C. Greubel, Feb 19 2021

Formula

T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.
Sum_{k=0..n} T(n, k, 3) = A000295(n) + Sum_{k=0..n} 3^(n*binomial(n-2, k-1)). - G. C. Greubel, Feb 19 2021

Extensions

Edited by G. C. Greubel, Feb 19 2021
Previous Showing 11-14 of 14 results.

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

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