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.

A174119 Triangle T(n, k) = ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k) with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 14, 70, 14, 1, 1, 30, 420, 420, 30, 1, 1, 55, 1650, 4620, 1650, 55, 1, 1, 91, 5005, 30030, 30030, 5005, 91, 1, 1, 140, 12740, 140140, 300300, 140140, 12740, 140, 1, 1, 204, 28560, 519792, 2042040, 2042040, 519792, 28560, 204, 1, 1, 285, 58140, 1627920, 10581480, 19399380, 10581480, 1627920, 58140, 285, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 08 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   1,     1;
  1,   5,     5,       1;
  1,  14,    70,      14,        1;
  1,  30,   420,     420,       30,        1;
  1,  55,  1650,    4620,     1650,       55,        1;
  1,  91,  5005,   30030,    30030,     5005,       91,       1;
  1, 140, 12740,  140140,   300300,   140140,    12740,     140,     1;
  1, 204, 28560,  519792,  2042040,  2042040,   519792,   28560,   204,   1;
  1, 285, 58140, 1627920, 10581480, 19399380, 10581480, 1627920, 58140, 285, 1;

Links

Crossrefs

Cf. A174116, A174117, A174124, A174125.
Cf. A111910, A196148.

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else ((n-k)/6)*Binomial(n-1, k-1)*Binomial(2*n, 2*k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021
  • Mathematica
    (* First program *)
c[n_]:= If[n<2, 1, Product[j*(j-1)*(2*j-1)/6, {j, 2, n}]];
T[n_, k_]:= c[n]/(c[k]*c[n-k]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, ((n-k)/6)*Binomial[n-1, k-1]*Binomial[2*n, 2*k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)
  • Sage
    def T(n,k): return 1 if (k==0 or k==n) else ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021

Formula

T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=2..n} j*(j-1)*(2*j-1)/6 for n > 2 otherwise 1.
From G. C. Greubel, Feb 11 2021: (Start)
T(n, k) = ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k) with T(n, 0) = T(n, n) =1.
Sum_{k=0..n} T(n, k) = (n*(n-1)*(2*n-1)/6)*HypergeometricPFQ[{1-n, 3/2-n, 2-n}, {3/2, 2}, -1] + 2 - [n=0] (n*(n-1)*(2*n-1)/6)*A196148[n-2] + 2 - [n=0]. (End)

Extensions

Edited by G. C. Greubel, Feb 11 2021

A174116 Triangle T(n, k) = (n/2)*binomial(n-1, k-1)*binomial(n-1, k) with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 18, 6, 1, 1, 10, 60, 60, 10, 1, 1, 15, 150, 300, 150, 15, 1, 1, 21, 315, 1050, 1050, 315, 21, 1, 1, 28, 588, 2940, 4900, 2940, 588, 28, 1, 1, 36, 1008, 7056, 17640, 17640, 7056, 1008, 36, 1, 1, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 08 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,    1;
  1,  3,    3,     1;
  1,  6,   18,     6,     1;
  1, 10,   60,    60,    10,     1;
  1, 15,  150,   300,   150,    15,     1;
  1, 21,  315,  1050,  1050,   315,    21,     1;
  1, 28,  588,  2940,  4900,  2940,   588,    28,    1;
  1, 36, 1008,  7056, 17640, 17640,  7056,  1008,   36,  1;
  1, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 1;

Links

Crossrefs

Cf. A174117, A174119, A174124, A174125.
Cf. A000108, A001263.

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else (n/2)*Binomial(n-1, k-1)*Binomial(n-1, k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021
  • Mathematica
    (* First program *)
c[n_]:= If[n<2, 1, Product[Binomial[j,2], {j, 2, n}]];
T[n_, k_]:= c[n]/(c[k]*c[n-k]);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, (n/2)*Binomial[n-1, k-1]*Binomial[n-1, k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)
  • Sage
    def T(n,k): return 1 if (k==0 or k==n) else (n/2)*binomial(n-1, k-1)*binomial(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021

Formula

Let c(n) = Product_{j=2..n} binomial(j,2) for n > 1 otherwise 1 then the number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).
From G. C. Greubel, Feb 11 2021: (Start)
T(n, k) = (n/2)*binomial(n-1, k-1)*binomial(n-1, k) with T(n, 0) = T(n, n) = 1.
T(n, k) = binomial(n-k+1, 2)*A001263(n, k) with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} T(n,k) = binomial(n, 2)*C_{n-1} + 2 - [n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. (End)

Extensions

Edited by G. C. Greubel, Feb 11 2021

A174117 Triangle T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 15, 40, 15, 1, 1, 24, 120, 120, 24, 1, 1, 35, 280, 525, 280, 35, 1, 1, 48, 560, 1680, 1680, 560, 48, 1, 1, 63, 1008, 4410, 7056, 4410, 1008, 63, 1, 1, 80, 1680, 10080, 23520, 23520, 10080, 1680, 80, 1, 1, 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 08 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  3,    1;
  1,  8,    8,     1;
  1, 15,   40,    15,     1;
  1, 24,  120,   120,    24,     1;
  1, 35,  280,   525,   280,    35,     1;
  1, 48,  560,  1680,  1680,   560,    48,     1;
  1, 63, 1008,  4410,  7056,  4410,  1008,    63,    1;
  1, 80, 1680, 10080, 23520, 23520, 10080,  1680,   80,  1;
  1, 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99, 1;

Links

Crossrefs

Cf. A174116, A174119, A174124, A174125.
Cf. A000108, A001263.

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else (2*k/(k+1))*Binomial(n-1, k)*Binomial(n+1, k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021
  • Mathematica
    (* First program *)
c[n_]:= If[n<2, 1, Product[i^2 -1, {i,2,n}]];
T[n_, k_]:= c[n]/(c[k]*c[n-k]);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, (2*k/(k+1))*Binomial[n+1, k]*Binomial[n-1, k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)
  • Sage
    def T(n,k): return 1 if (k==0 or k==n) else (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021

Formula

Let c(n) = Product_{j=2..n} (j^2 - 1) for n > 1 otherwise 1 then the number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).
From G. C. Greubel, Feb 11 2021: (Start)
T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1.
T(n, k) = 2*((n+1)*(n-k)/(k+1))*A001263(n, k).
Sum_{k=0..n} T(n, k) = (2/(n+2))*( (n^2-1)*C_{n} + 1), where C_{n} are the Catalan numbers (A000108). (End)

Extensions

Edited by G. C. Greubel, Feb 11 2021

A174125 Triangle T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 24, 45, 24, 1, 1, 35, 105, 105, 35, 1, 1, 48, 210, 336, 210, 48, 1, 1, 63, 378, 882, 882, 378, 63, 1, 1, 80, 630, 2016, 2940, 2016, 630, 80, 1, 1, 99, 990, 4158, 8316, 8316, 4158, 990, 99, 1, 1, 120, 1485, 7920, 20790, 28512, 20790, 7920, 1485, 120, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 09 2010

Keywords

Comments

Triangles of this class, depending upon q, are of the form T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and have the row sums Sum_{k=0..n} T(n, k, q) = q*(q+1)*C_{n+q}/binomial(n+2*q, q-1) - 2*q + q*[n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. - G. C. Greubel, Feb 11 2021

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   8,    1;
  1,  15,   15,    1;
  1,  24,   45,   24,     1;
  1,  35,  105,  105,    35,     1;
  1,  48,  210,  336,   210,    48,     1;
  1,  63,  378,  882,   882,   378,    63,    1;
  1,  80,  630, 2016,  2940,  2016,   630,   80,    1;
  1,  99,  990, 4158,  8316,  8316,  4158,  990,   99,   1;
  1, 120, 1485, 7920, 20790, 28512, 20790, 7920, 1485, 120, 1;

Links

Crossrefs

Cf. A174124 (q=1), this sequence (q=2).
Cf. A000108, A174116, A174117, A174119.

Programs

  • Magma
    c:= func< n,q | n lt 2 select 1 else (&*[j*(j+q): j in [2..n]]) >;
T:= func< n,k,q | c(n, q)/(c(k, q)*c(n-k, q)) >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021
  • Mathematica
    (* First program *)
c[n_, q_]:= If[n<2, 1, Product[i*(i+q), {i,2,n}]];
T[n_, m_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_, q_]:= If[k==0 || k==n, 1, (q+1)*Binomial[n, k]*(Pochhammer[q+1, n]/(Pochhammer[q+1, k]*Pochhammer[q+1, n-k]))];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)
  • Sage
    def T(n,k,q): return 1 if (k==0 or k==n) else (q+1)*binomial(n, k)*(rising_factorial(q+1, n)/(rising_factorial(q+1, k)*rising_factorial(q+1, n-k)))
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021

Formula

Let c(n, q) = Product_{j=2..n} j*(j+q) for n > 2, otherwise 1, then the number triangle is given by T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) for q = 2.
From G. C. Greubel, Feb 11 2021: (Start)
T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 2.
Sum_{k=0..n} T(n, k, 1) = 6*A000108(n+2)/(n+4) - 4 + 2*[n=0]. (End)

Extensions

Edited by G. C. Greubel, Feb 11 2021
Showing 1-4 of 4 results.

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