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 1-3 of 3 results.

A172364 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 10, 30, 30, 10, 1, 1, 31, 310, 930, 310, 31, 1, 1, 94, 2914, 29140, 29140, 2914, 94, 1, 1, 285, 26790, 830490, 2768300, 830490, 26790, 285, 1, 1, 865, 246525, 23173350, 239457950, 239457950, 23173350, 246525, 865, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 01 2010

Keywords

Comments

Let f be the sequence 0, 1, 1, 1, 3, 10, 31, 94, 285, 865, 2626, 7972, 24201.., f(n) = 3*f(n-1)+f(n-4), and c the partial products of f: c(n) = 1, 1, 1, 1, 3, 30, 930, 87420, 24914700, 21551215500, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   1,      1;
  1,   1,      1,        1;
  1,   3,      3,        3,         1;
  1,  10,     30,       30,        10,         1;
  1,  31,    310,      930,       310,        31,        1;
  1,  94,   2914,    29140,     29140,      2914,       94,      1;
  1, 285,  26790,   830490,   2768300,    830490,    26790,    285,   1;
  1, 865, 246525, 23173350, 239457950, 239457950, 23173350, 246525, 865, 1;

Links

Crossrefs

Cf. A172363 (q=1), this sequence (q=3).

Programs

  • Mathematica
    f[n_, q_]:= f[n, q]= If[n==0,0,If[n<4, 1, q*f[n-1, q] + f[n-4, q]]];
c[n_, q_]:= Product[f[j, q], {j,n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 08 2021 *)
  • Sage
    @CachedFunction
def f(n,q): return 0 if (n==0) else 1 if (n<4) else q*f(n-1, q) + f(n-4, q)
def c(n,q): return product( f(j,q) for j in (1..n) )
def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021

Formula

T(n, k, q) = round( c(n,q)/(c(k,q)*c(n-k,q)) ), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*f(n-1, q) + f(n-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 3. - G. C. Greubel, May 08 2021

Extensions

Definition corrected to give integral terms, G. C. Greubel, May 08 2021

A172368 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c is a sequence defined in comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 5, 15, 15, 15, 5, 1, 1, 7, 35, 105, 105, 35, 7, 1, 1, 9, 63, 315, 945, 315, 63, 9, 1, 1, 15, 135, 945, 4725, 4725, 945, 135, 15, 1, 1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 01 2010

Keywords

Comments

Start from A052942 and its partial products c(n) = 1, 1, 1, 1, 1, 3, 15, 105, 945, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,   1;
  1,  1,   1,    1;
  1,  1,   1,    1,     1;
  1,  3,   3,    3,     3,     1;
  1,  5,  15,   15,    15,     5,     1;
  1,  7,  35,  105,   105,    35,     7,    1;
  1,  9,  63,  315,   945,   315,    63,    9,   1;
  1, 15, 135,  945,  4725,  4725,   945,  135,  15,  1;
  1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1;

Links

Crossrefs

Cf. A052942, A172363 (q=1), this sequence (q=2), A172369 (q=3).

Programs

  • Mathematica
    f[n_, q_]:= f[n, q]= If[n==0,0,If[n<4, 1, f[n-1, q] + q*f[n-4, q]]];
c[n_, q_]:= Product[f[j, q], {j,n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 08 2021 *)
  • Sage
    @CachedFunction
def f(n,q): return 0 if (n==0) else 1 if (n<4) else f(n-1, q) + q*f(n-4, q)
def c(n,q): return product( f(j,q) for j in (1..n) )
def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021

Formula

T(n, k, q) = round( c(n,q)/(c(k,q)*c(n-k,q)) ), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = f(n-1, q) + q*f(n-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 2. - G. C. Greubel, May 08 2021

Extensions

Definition corrected to give integral terms, G. C. Greubel, May 08 2021

A172369 Triangle read by rows: T(n,k,q) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 1, 1, 7, 28, 28, 28, 7, 1, 1, 10, 70, 280, 280, 70, 10, 1, 1, 13, 130, 910, 3640, 910, 130, 13, 1, 1, 25, 325, 3250, 22750, 22750, 3250, 325, 25, 1, 1, 46, 1150, 14950, 149500, 261625, 149500, 14950, 1150, 46, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 01 2010

Keywords

Comments

Start from A143454 and its partial products c(n) = 1, 1, 1, 1, 1, 4, 28, 280, 3640, 91000, 4186000, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,    1;
  1,  1,    1,     1;
  1,  1,    1,     1,      1;
  1,  4,    4,     4,      4,      1;
  1,  7,   28,    28,     28,      7,      1;
  1, 10,   70,   280,    280,     70,     10,     1;
  1, 13,  130,   910,   3640,    910,    130,    13,    1;
  1, 25,  325,  3250,  22750,  22750,   3250,   325,   25,  1;
  1, 46, 1150, 14950, 149500, 261625, 149500, 14950, 1150, 46, 1;

Links

Crossrefs

Cf. A172363 (q=1), A172368 (q=2), this sequence (q=3), A318774.

Programs

  • Mathematica
    f[n_, q_]:= f[n, q]= If[n==0, 0, If[n<4, 1, f[n-1, q] +q*f[n-4, q]]];
c[n_, q_]:= Product[f[j, q], {j,n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n - k, q])];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 08 2021 *)
  • Sage
    @CachedFunction
def f(n,q): return 0 if (n==0) else 1 if (n<4) else f(n-1, q) + q*f(n-4, q)
def c(n,q): return product( f(j,q) for j in (1..n) )
def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021

Formula

T(n, k, q) = round( c(n,q)/(c(k,q)*c(n-k,q)) ), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = f(n-1, q) + q*f(n-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 3. - G. C. Greubel, May 08 2021

Extensions

Definition corrected to give integral terms, G. C. Greubel, May 08 2021
Showing 1-3 of 3 results.

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