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 1-5 of 5 results.

A155863 Triangle T(n,k) = n*(n^2 - 1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 24, 24, 1, 1, 60, 120, 60, 1, 1, 120, 360, 360, 120, 1, 1, 210, 840, 1260, 840, 210, 1, 1, 336, 1680, 3360, 3360, 1680, 336, 1, 1, 504, 3024, 7560, 10080, 7560, 3024, 504, 1, 1, 720, 5040, 15120, 25200, 25200, 15120, 5040, 720, 1, 1, 990, 7920, 27720, 55440, 69300, 55440, 27720, 7920, 990, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 29 2009

Keywords

Examples

			Triangle begins:
  1;
  1,   1;
  1,   6,    1;
  1,  24,   24,     1;
  1,  60,  120,    60,     1;
  1, 120,  360,   360,   120,     1;
  1, 210,  840,  1260,   840,   210,     1;
  1, 336, 1680,  3360,  3360,  1680,   336,     1;
  1, 504, 3024,  7560, 10080,  7560,  3024,   504,    1,
  1, 720, 5040, 15120, 25200, 25200, 15120,  5040,  720,   1;
  1, 990, 7920, 27720, 55440, 69300, 55440, 27720, 7920, 990, 1;
  ...

Links

Crossrefs

Cf. A001789, A052771, A155864, A155865.

Programs

  • Magma
    A155863:= func< n,k | k eq 0 or k eq n select 1 else 6*Binomial(n+1, 3)*Binomial(n-2, k-1) >;
[A155863(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021
  • Mathematica
    (* First program *)
p[n_, x_]:= p[n, x]= If[n==0, 1, 1 + x^n + x*D[(x+1)^(n+1), {x, 3}]];
Flatten[Table[CoefficientList[p[n,x], x], {n,0,12}]]
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, 6*Binomial[n+1, 3]*Binomial[n-2, k-1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2021 *)
  • Maxima
    T(n, k):= ratcoef(expand(x^n + n*(n^2 -1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2), x, k)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 03 2018 */
  • Sage
    def A155863(n,k): return 1 if (k==0 or k==n) else 6*binomial(n+1, 3)*binomial(n-2, k-1)
flatten([[A155863(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

Formula

T(n, k) = coefficients of p(n, x), where p(n, x) = 1 + x^n + x*((d/dx)^3 (x+1)^(n+1)) and T(0, 0) = 1.
From Franck Maminirina Ramaharo, Dec 03 2018: (Start)
T(n, k) = (n-1)*n*(n+1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.
n-th row polynomial is x^n + n*(n^2 - 1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2.
G.f.: 1/(1 - y) + 1/(1 - x*y) + (6*x*y^2)/(1 - y - x*y)^4 - 1.
E.g.f.: exp(y) + exp(x*y) + (3*x*y^2 + (x + x^2)*y^3)*exp((1 + x)*y) - 1. (End)
Sum_{k=0..n} T(n, k) = 2 - [n=0] + 6*A001789(n+1) = 2 - [n=0] + A052771(n+1). - G. C. Greubel, Jun 04 2021

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 03 2018

A155864 Triangle T(n,k) = n*(n-1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 24, 12, 1, 1, 20, 60, 60, 20, 1, 1, 30, 120, 180, 120, 30, 1, 1, 42, 210, 420, 420, 210, 42, 1, 1, 56, 336, 840, 1120, 840, 336, 56, 1, 1, 72, 504, 1512, 2520, 2520, 1512, 504, 72, 1, 1, 90, 720, 2520, 5040, 6300, 5040, 2520, 720, 90, 1
Offset: 0

Views

Author

Roger L. Bagula, Jan 29 2009

Keywords

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  6,   6,    1;
  1, 12,  24,   12,    1;
  1, 20,  60,   60,   20,    1;
  1, 30, 120,  180,  120,   30,    1;
  1, 42, 210,  420,  420,  210,   42,    1;
  1, 56, 336,  840, 1120,  840,  336,   56,   1;
  1, 72, 504, 1512, 2520, 2520, 1512,  504,  72,  1;
  1, 90, 720, 2520, 5040, 6300, 5040, 2520, 720, 90, 1;
  ...

Links

Crossrefs

Cf. A001815, A155863, A155865.

Programs

  • Magma
    A155864:= func< n,k | k eq 0 or k eq n select 1 else n*(n-1)*Binomial(n-2, k-1) >;
[A155864(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021
  • Mathematica
    (* First program *)
p[n_, x_]:= p[n,x]= If[n==0, 1, 1 + x^n + x*D[(x+1)^(n), {x, 2}]];
Flatten[Table[CoefficientList[p[n,x], x], {n, 0, 12}]]
(* Second program *)
Table[If[k==0 || k==n, 1, 2*Binomial[n, 2]*Binomial[n-2, k-1]], {n,0,12}, {k,0,n}] //Flatten (* G. C. Greubel, Jun 04 2021 *)
  • Maxima
    T(n, k) := ratcoef(x^n + n*(n-1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2, x, k)$ create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 04 2018 */
  • Sage
    def A155864(n,k): return 1 if (k==0 or k==n) else n*(n-1)*binomial(n-2,k-1)
flatten([[A155864(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

Formula

T(n, k) = coefficients of p(n, x), where p(n, x) = 1 + x^n + x*((d/dx)^2 (1+x)^n), with T(0, 0) = 1.
From Franck Maminirina Ramaharo, Dec 04 2018: (Start)
T(n, k) = n*(n-1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.
n-th row polynomial is x^n + n*(n - 1)*x*(x + 1)^(n - 2) + (1 + (-1)^(2^n))/2.
G.f.: 1/(1 - y) + 1/(1 - x*y) + 2*x*y^2/(1 - y - x*y)^3 - 1.
E.g.f.: exp(y) + exp(x*y) + x*y^2*exp(y + x*y) - 1. (End)
Sum_{k=0..n} T(n, k) = 2 - [n=0] + A001815(n). - G. C. Greubel, Jun 04 2021

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 04 2018

A174126 Triangle T(n, k) = (n-k)^2 * binomial(n-1, k-1)^2 with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 9, 36, 9, 1, 1, 16, 144, 144, 16, 1, 1, 25, 400, 900, 400, 25, 1, 1, 36, 900, 3600, 3600, 900, 36, 1, 1, 49, 1764, 11025, 19600, 11025, 1764, 49, 1, 1, 64, 3136, 28224, 78400, 78400, 28224, 3136, 64, 1, 1, 81, 5184, 63504, 254016, 396900, 254016, 63504, 5184, 81, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 09 2010

Keywords

Comments

This triangle sequence is part of a class of triangles defined by T(n, k, q) = (n-k)^q * binomial(n-1, k-1)^q with T(n, 0) = T(n, n) = 1 and have row sums Sum_{k=0..n} T(n, k, q) = 2 - [n=0] + Sum_{k=1..n-1} k^q * binomial(n-1, k)^q. - G. C. Greubel, Feb 10 2021

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,    1;
  1,  4,    4,     1;
  1,  9,   36,     9,      1;
  1, 16,  144,   144,     16,      1;
  1, 25,  400,   900,    400,     25,      1;
  1, 36,  900,  3600,   3600,    900,     36,     1;
  1, 49, 1764, 11025,  19600,  11025,   1764,    49,    1;
  1, 64, 3136, 28224,  78400,  78400,  28224,  3136,   64,  1;
  1, 81, 5184, 63504, 254016, 396900, 254016, 63504, 5184, 81, 1;

Links

Crossrefs

Cf. A000108, A037966.
Cf. A155865 (q=1), this sequence (q=2), A174127 (q=3).

Programs

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

Formula

Let c(n) = Product_{i=2..n} (i-1)^2 for n > 2 otherwise 1. The number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).
From G. C. Greubel, Feb 10 2021: (Start)
T(n, k) = (n-k)^2 * binomial(n-1, k-1)^2 with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} T(n, k) = 2 + A037966(n-1) - [n=0] = 2 + (n-1)^3*C_{n-2} - [n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. (End)

Extensions

Edited by G. C. Greubel, Feb 10 2021

A174127 Triangle T(n, k) = (n-k)^3 * binomial(n-1, k-1)^3 with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 8, 1, 1, 27, 216, 27, 1, 1, 64, 1728, 1728, 64, 1, 1, 125, 8000, 27000, 8000, 125, 1, 1, 216, 27000, 216000, 216000, 27000, 216, 1, 1, 343, 74088, 1157625, 2744000, 1157625, 74088, 343, 1, 1, 512, 175616, 4741632, 21952000, 21952000, 4741632, 175616, 512, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 09 2010

Keywords

Comments

This triangle sequence is part of a class of triangles defined by T(n, k, q) = (n-k)^q * binomial(n-1, k-1)^q with T(n, 0) = T(n, n) = 1 and have row sums Sum_{k=0..n} T(n, k, q) = 2 - [n=0] + Sum_{k=1..n-1} k^q * binomial(n-1, k)^q. - G. C. Greubel, Feb 11 2021

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   1,      1;
  1,   8,      8,       1;
  1,  27,    216,      27,        1;
  1,  64,   1728,    1728,       64,        1;
  1, 125,   8000,   27000,     8000,      125,       1;
  1, 216,  27000,  216000,   216000,    27000,     216,      1;
  1, 343,  74088, 1157625,  2744000,  1157625,   74088,    343,   1;
  1, 512, 175616, 4741632, 21952000, 21952000, 4741632, 175616, 512, 1;

Links

Crossrefs

Cf. A155865 (q=1), A174126 (q=2), this sequence (q=3).
Cf. A000172.

Programs

  • Magma
    T:= func< n,k,q | k eq 0 or k eq n select 1 else (n-k)^q*Binomial(n-1,k-1)^q >;
[T(n,k,3): 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-1)^3, {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_, q_]:= If[k==0 || k==n, 1, (n-k)^q*Binomial[n-1, k-1]^q];
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 10 2021 *)
  • Sage
    def T(n,k,q): return 1 if (k==0 or k==n) else (n-k)^q*binomial(n-1,k-1)^q
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021

Formula

Let c(n) = Product_{i=2..n} (i-1)^3 for n > 2 otherwise 1. 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-k)^3 * binomial(n-1, k-1)^3 with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} T(n, k) = 2 + (n-1)^3*A000172(n-2) - [n=0]. (End)

Extensions

Edited by G. C. Greubel, Feb 11 2021

A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 28, 42, 28, 1, 1, 125, 250, 250, 125, 1, 1, 726, 1815, 2420, 1815, 726, 1, 1, 5047, 15141, 25235, 25235, 15141, 5047, 1, 1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1, 1, 362889, 1451556, 3386964, 5080446, 5080446, 3386964, 1451556, 362889, 1
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Keywords

Comments

Row 0 is 1, and row n gives the coefficients in the expansion of (x + 1)^n + (n - 1)!*((x + 1)^n - x^n -1). - Franck Maminirina Ramaharo, Dec 22 2018

Examples

			Triangle begins:
  1;
  1,     1;
  1,     4,      1;
  1,     9,      9,      1;
  1,    28,     42,     28,      1;
  1,   125,    250,    250,    125,      1;
  1,   726,   1815,   2420,   1815,    726,      1;
  1,  5047,  15141,  25235,  25235,  15141,   5047,     1;
  1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1;
  ...

Crossrefs

Row sums: A154252.
Cf. A007318, A132047, A155863, A155864, A155865 A168619, A219570.

Programs

  • Mathematica
    p[x_, n_] := If[n == 0, 1, (x + 1)^n + (n - 1)!*((x + 1)^n - x^n - 1)];
Table[CoefficientList[p[x, n], x], {n, 0, 12}] // Flatten (* Franck Maminirina Ramaharo, Dec 22 2018 *)
  • Maxima
    T(n, k) := if k = 0 or k = n then 1 else ((n - 1)! + 1)*binomial(n, k)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 22 2018 */

Formula

From Franck Maminirina Ramaharo, Dec 22 2018: (Start)
T(n,k) = A007318(n,k) + A219570(n,k) for 1 <= k <= n - 1, n >= 2.
E.g.f.: exp((1 + x)*y) + log((1 - y)*(1 - x*y)/(1 - (1 + x)*y)). (End)

Extensions

Edited by Franck Maminirina Ramaharo, Dec 22 2018 (previous definition and examples were the same as A168620, but with different entries, as pointed out by R. J. Mathar, Oct 21 2012)
Showing 1-5 of 5 results.

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