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.

A132736 Row sums of triangle A132735.

Original entry on oeis.org

1, 2, 5, 10, 19, 36, 69, 134, 263, 520, 1033, 2058, 4107, 8204, 16397, 32782, 65551, 131088, 262161, 524306, 1048595, 2097172, 4194325, 8388630, 16777239, 33554456, 67108889, 134217754, 268435483, 536870940, 1073741853, 2147483678
Offset: 0

Views

Author

Gary W. Adamson, Aug 26 2007

Keywords

Comments

Apart from first term, the same as A052944. - R. J. Mathar, Jun 12 2008

Examples

			a(4) = 19 = sum of row 4 terms of triangle A132735: (1 + 5 + 7 + 5 + 1).
a(3) = 10 = (1, 3, 3, 1) dot (1, 1, 2, 0) = (1 + 3 + 6 + 0).

Links

Crossrefs

Cf. A132735.

Programs

  • Magma
    [1] cat [2^n + n-1: n in [1..30]]; // G. C. Greubel, Feb 14 2021
  • Maple
    a:= proc(n) option remember; if n=0 then 1 else add((binomial(n,j)+1), j=0..n-1) fi end: seq(a(n), n=0..31);# Zerinvary Lajos, Mar 29 2009
  • Mathematica
    Table[2^n + n-1 + Boole[n==0], {n,0,30}] (* G. C. Greubel, Feb 14 2021 *)
  • Sage
    [1]+[2^n + n-1 for n in (1..30)] # G. C. Greubel, Feb 14 2021

Formula

Binomial transform of [1, 1, 2, 0, 2, 0, 2, 0, 2, ...].
From Colin Barker, Aug 12 2012: (Start)
a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3) for n>3.
G.f.: (1 -2*x +2*x^2 -2*x^3)/((1-x)^2*(1-2*x)). (End)
From G. C. Greubel, Feb 14 2021: (Start)
a(n) = 2^n + n - 1 + [n=0].
E.g.f.: 1 - (1-x)*exp(x) + exp(2*x). (End)

Extensions

More terms from R. J. Mathar, Jun 12 2008

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

A173741 T(n,k) = binomial(n,k) + 4 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 7, 7, 1, 1, 8, 10, 8, 1, 1, 9, 14, 14, 9, 1, 1, 10, 19, 24, 19, 10, 1, 1, 11, 25, 39, 39, 25, 11, 1, 1, 12, 32, 60, 74, 60, 32, 12, 1, 1, 13, 40, 88, 130, 130, 88, 40, 13, 1, 1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1, 1, 15, 59, 169, 334, 466, 466, 334, 169, 59, 15, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 23 2010

Keywords

Comments

For n >= 1, row n sums to 2*A100314(n).

Examples

			Triangle begins:
  1;
  1,  1;
  1,  6,  1;
  1,  7,  7,   1;
  1,  8, 10,   8,   1;
  1,  9, 14,  14,   9,   1;
  1, 10, 19,  24,  19,  10,   1;
  1, 11, 25,  39,  39,  25,  11,   1;
  1, 12, 32,  60,  74,  60,  32,  12,  1;
  1, 13, 40,  88, 130, 130,  88,  40, 13,  1;
  1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1;
  ...

Links

Crossrefs

Cf. A100314, A103451, A156050.
Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), A173740 (q=2), this sequence (q=4), A173742 (q=6).

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 4 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021
  • Mathematica
    T[n_, m_] = Binomial[n, m] + 4*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) + 4$
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) + 4
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) + 2*(1 - A103451(n,k)).
T(n,k) = 5*A007318(n,k) - 4*A132044(n,k).
n-th row polynomial is 2*(1 - (-1)^(2^n)) + (1 + x)^n + 4*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 5*x*y^2 - 4*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (4 - 4*x + 4*x*exp(y) - 4*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 4*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018

A132737 Triangle T(n,k) = 2*binomial(n,k) + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 7, 7, 1, 1, 9, 13, 9, 1, 1, 11, 21, 21, 11, 1, 1, 13, 31, 41, 31, 13, 1, 1, 15, 43, 71, 71, 43, 15, 1, 1, 17, 57, 113, 141, 113, 57, 17, 1, 1, 19, 73, 169, 253, 253, 169, 73, 19, 1, 1, 21, 91, 241, 421, 505, 421, 241, 91, 21, 1, 1, 23, 111, 331, 661, 925, 925, 661, 331, 111, 23, 1
Offset: 0

Views

Author

Gary W. Adamson, Aug 26 2007

Keywords

Examples

			First few rows of the triangle are:
  1;
  1,  1;
  1,  5,  1;
  1,  7,  7,  1;
  1,  9, 13,  9,  1;
  1, 11, 21, 21, 11,  1;
  1, 13, 31, 41, 31, 13,  1;
  1, 15, 43, 71, 71, 43, 15, 1;
  ...

Links

Crossrefs

Cf. A132735, A132738.
Sequences of the form 2*binomial(n,k) + q: A132729 (q=-3), A132731 (q=-2), A109128 (q=-1), A132046 (q=0), this sequence (q=1).

Programs

  • Magma
    A132737:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) +1 >;
[A132737(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 15 2021
  • Mathematica
    T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n,k] +1];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)
  • Sage
    def A132737(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) + 1
flatten([[A132737(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 15 2021

Formula

T(n, k) = 2*A132735(n, k) - 1, an infinite lower triangular matrix.
T(n,0) = T(n,n) = 1; otherwise T(n,k) = 2*C(n,k) + 1. - Franklin T. Adams-Watters, Jul 06 2009
Sum_{k=0..n} T(n, k) = 2^(n+1) + n - 3 + 2*[n=0] = A132738(n). - G. C. Greubel, Feb 15 2021

Extensions

Extended by Franklin T. Adams-Watters, Jul 06 2009
Showing 1-4 of 4 results.

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