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.

Showing 1-4 of 4 results.

A173117 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 8, 5, 1, 1, 6, 14, 14, 6, 1, 1, 7, 20, 28, 20, 7, 1, 1, 8, 27, 49, 49, 27, 8, 1, 1, 9, 35, 76, 98, 76, 35, 9, 1, 1, 10, 44, 111, 175, 175, 111, 44, 10, 1, 1, 11, 54, 155, 286, 350, 286, 155, 54, 11, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 10 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  4,   1;
  1,  5,  8,   5,   1;
  1,  6, 14,  14,   6,   1;
  1,  7, 20,  28,  20,   7,   1;
  1,  8, 27,  49,  49,  27,   8,   1;
  1,  9, 35,  76,  98,  76,  35,   9,  1;
  1, 10, 44, 111, 175, 175, 111,  44, 10,  1;
  1, 11, 54, 155, 286, 350, 286, 155, 54, 11, 1;

Links

Crossrefs

Cf. A007318 (q=0), A072405 (q= -1), this sequence (q=1), A173118 (q=2), A173119 (q=3), A173120 (q= -4), A173122.

Programs

  • Mathematica
    T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j,0,5}]];
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)
  • Sage
    @CachedFunction
def T(n,k,q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021

Formula

T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 1.
Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = 1. - G. C. Greubel, Apr 27 2021

Extensions

Edited by G. C. Greubel, Apr 27 2021

A173118 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 10, 6, 1, 1, 7, 20, 20, 7, 1, 1, 8, 27, 40, 27, 8, 1, 1, 9, 35, 75, 75, 35, 9, 1, 1, 10, 44, 110, 150, 110, 44, 10, 1, 1, 11, 54, 154, 276, 276, 154, 54, 11, 1, 1, 12, 65, 208, 430, 552, 430, 208, 65, 12, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 10 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  4,  1;
  1,  5,  5,   1;
  1,  6, 10,   6,   1;
  1,  7, 20,  20,   7,   1;
  1,  8, 27,  40,  27,   8,   1;
  1,  9, 35,  75,  75,  35,   9,   1;
  1, 10, 44, 110, 150, 110,  44,  10,  1;
  1, 11, 54, 154, 276, 276, 154,  54, 11,  1;
  1, 12, 65, 208, 430, 552, 430, 208, 65, 12, 1;

Links

Crossrefs

Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), this sequence (q=2), A173119 (q=3), A173120 (q= -4), A173122.

Programs

  • Mathematica
    T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j,0,5}]];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)
  • Sage
    @CachedFunction
def T(n,k,q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021

Formula

T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 2.
Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = 2. - G. C. Greubel, Apr 27 2021

Extensions

Edited by G. C. Greubel, Apr 27 2021

A173119 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 6, 6, 1, 1, 7, 12, 7, 1, 1, 8, 28, 28, 8, 1, 1, 9, 36, 56, 36, 9, 1, 1, 10, 45, 119, 119, 45, 10, 1, 1, 11, 55, 164, 238, 164, 55, 11, 1, 1, 12, 66, 219, 483, 483, 219, 66, 12, 1, 1, 13, 78, 285, 702, 966, 702, 285, 78, 13, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 10 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  5,  1;
  1,  6,  6,   1;
  1,  7, 12,   7,   1;
  1,  8, 28,  28,   8,   1;
  1,  9, 36,  56,  36,   9,   1;
  1, 10, 45, 119, 119,  45,  10,   1;
  1, 11, 55, 164, 238, 164,  55,  11,  1;
  1, 12, 66, 219, 483, 483, 219,  66, 12,  1;
  1, 13, 78, 285, 702, 966, 702, 285, 78, 13, 1;

Links

Crossrefs

Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), this sequence (q=3), A173120 (q= -4), A173122.

Programs

  • Mathematica
    T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j,0,5}]];
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)
  • Sage
    @CachedFunction
def T(n,k,q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
flatten([[T(n,k,3) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Apr 27 2021

Formula

T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3.
Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = 3. - G. C. Greubel, Apr 27 2021

Extensions

Edited by G. C. Greubel, Apr 27 2021

A173120 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = -4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, 0, -2, 0, 1, 1, 1, 14, 14, 1, 1, 1, 2, 15, 28, 15, 2, 1, 1, 3, 17, -21, -21, 17, 3, 1, 1, 4, 20, -4, -42, -4, 20, 4, 1, 1, 5, 24, 16, 210, 210, 16, 24, 5, 1, 1, 6, 29, 40, 226, 420, 226, 40, 29, 6, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 10 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1, -2,  1;
  1, -1, -1,   1;
  1,  0, -2,   0,   1;
  1,  1, 14,  14,   1,   1;
  1,  2, 15,  28,  15,   2,   1;
  1,  3, 17, -21, -21,  17,   3,  1;
  1,  4, 20,  -4, -42,  -4,  20,  4,  1;
  1,  5, 24,  16, 210, 210,  16, 24,  5, 1;
  1,  6, 29,  40, 226, 420, 226, 40, 29, 6, 1;

Links

Crossrefs

Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), A173119 (q=3), this sequence (q= -4), A173122.

Programs

  • Mathematica
    T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j,0,5}]];
Table[T[n,k,-4], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)
  • Sage
    @CachedFunction
def T(n,k,q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
flatten([[T(n,k,-4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021

Formula

T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = -4.
Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = -4. - G. C. Greubel, Apr 27 2021

Extensions

Edited by G. C. Greubel, Apr 27 2021
Showing 1-4 of 4 results.

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

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