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.

A132735 Triangle T(n,k) = binomial(n,k) + 1 with T(n,0) = T(n,n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 7, 5, 1, 1, 6, 11, 11, 6, 1, 1, 7, 16, 21, 16, 7, 1, 1, 8, 22, 36, 36, 22, 8, 1, 1, 9, 29, 57, 71, 57, 29, 9, 1, 1, 10, 37, 85, 127, 127, 85, 37, 10, 1, 1, 11, 46, 121, 211, 253, 211, 121, 46, 11, 1, 1, 12, 56, 166, 331, 463, 463, 331, 166, 56, 12, 1
Offset: 0

Views

Author

Gary W. Adamson, Aug 26 2007

Keywords

Examples

			First few rows of the triangle are:
  1;
  1, 1;
  1, 3,  1;
  1, 4,  4,  1;
  1, 5,  7,  5,  1;
  1, 6, 11, 11,  6, 1;
  1, 7, 16, 21, 16, 7, 1;
  ...

Links

Crossrefs

Cf. A103451, A132736.
Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), this sequence (q=1), A173740 (q=2), A173741 (q=4), A173742 (q=6).

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 1 >;
[T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021
  • Mathematica
    T[n_, k_]:= If[k==0||k==n, 1, Binomial[n,k] +1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)
  • Sage
    def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 1
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021

Formula

T(n, k) = A007318(n,k) + 1 - A103451(n,k), an infinite lower triangular matrix.
T(n,0) = T(n,n) = 1; T(n,k) = C(n,k) + 1 otherwise. - Franklin T. Adams-Watters, Jul 06 2009
Sum_{k=0..n} T(n, k) = 2^n + n - 1 + [n=0] = A132736(n). - G. C. Greubel, Feb 14 2021

Extensions

Corrected and extended by Franklin T. Adams-Watters, Jul 06 2009

A173740 Triangle T(n,k) = binomial(n,k) + 2 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, 4, 1, 1, 5, 5, 1, 1, 6, 8, 6, 1, 1, 7, 12, 12, 7, 1, 1, 8, 17, 22, 17, 8, 1, 1, 9, 23, 37, 37, 23, 9, 1, 1, 10, 30, 58, 72, 58, 30, 10, 1, 1, 11, 38, 86, 128, 128, 86, 38, 11, 1, 1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1, 1, 13, 57, 167, 332, 464, 464, 332, 167, 57, 13, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 23 2010

Keywords

Comments

For n >= 1, row n sums to A131520(n).

Examples

			Triangle begins:
  1;
  1,  1;
  1,  4,  1;
  1,  5,  5,   1;
  1,  6,  8,   6,   1;
  1,  7, 12,  12,   7,   1;
  1,  8, 17,  22,  17,   8,   1;
  1,  9, 23,  37,  37,  23,   9,   1;
  1, 10, 30,  58,  72,  58,  30,  10,  1;
  1, 11, 38,  86, 128, 128,  86,  38, 11,  1;
  1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1;
  ...

Links

Crossrefs

Cf. A100314, A103451, A131520, A156050.
Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), this sequence (q=2), A173741 (q=4), A173742 (q=6).

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 2 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021
  • Mathematica
    T[n_, m_] = Binomial[n, m] + 2*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]
  • Maxima
    T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 2$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 08 2018 */
  • Sage
    def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 2
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

Formula

From Franck Maminirina Ramaharo, Dec 08 2018:(Start)
T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).
T(n,k) = 3*A007318(n,k) - 2*A132044(n,k).
n-th row polynomial is 1 - (-1)^(2^n) + (1 + x)^n + 2*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 3*x*y^2 - 2*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (2 - 2*x + 2*x*exp(y) - 2*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 2*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 08 2018

A173742 Triangle T(n,k) = binomial(n,k) + 6 with T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 9, 9, 1, 1, 10, 12, 10, 1, 1, 11, 16, 16, 11, 1, 1, 12, 21, 26, 21, 12, 1, 1, 13, 27, 41, 41, 27, 13, 1, 1, 14, 34, 62, 76, 62, 34, 14, 1, 1, 15, 42, 90, 132, 132, 90, 42, 15, 1, 1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1, 1, 17, 61, 171, 336, 468, 468, 336, 171, 61, 17, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 23 2010

Keywords

Comments

For n >= 1, row n sums to A131520(n) + A008586(n).

Examples

			Triangle begins:
  1;
  1,  1;
  1,  8,  1;
  1,  9,  9,   1;
  1, 10, 12,  10,   1;
  1, 11, 16,  16,  11,   1;
  1, 12, 21,  26,  21,  12,   1;
  1, 13, 27,  41,  41,  27,  13,   1;
  1, 14, 34,  62,  76,  62,  34,  14,  1;
  1, 15, 42,  90, 132, 132,  90,  42, 15,  1;
  1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1;
  ...

Links

Crossrefs

Cf. A007318, A103451, A132044, A156050, A173740, A173741.

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) +6 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021
  • Mathematica
    T[n_, m_] = Binomial[n, m] + 6*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]
  • Maxima
    T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 6$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 09 2018 */
  • Sage
    def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 6
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

Formula

From Franck Maminirina Ramaharo, Dec 09 2018: (Start)
T(n,k) = A007318(n,k) + 6*(1 - A103451(n,k)).
T(n,k) = 7*A007318(n,k) - 6*A132044(n,k).
n-th row polynomial is 3*(1 - (-1)^(2^n)) + (1 + x)^n + 6*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 7*x*y^2 - 6*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (6 - 6*x + 6*x*exp(y) - 6*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 6*n - 6 + 6*[n=0]. - G. C. Greubel, Feb 13 2021

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018
Showing 1-3 of 3 results.

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