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 1-6 of 6 results.

A143333 Pascal's triangle binomial(n,m) read by rows, all even elements replaced by zero.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 0, 0, 1, 1, 5, 0, 0, 5, 1, 1, 0, 15, 0, 15, 0, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 9, 0, 0, 0, 0, 0, 0, 9, 1, 1, 0, 45, 0, 0, 0, 0, 0, 45, 0, 1
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 21 2008

Keywords

Comments

Row sums are A088560.
A047999(n,k) = A057427(T(n,k)). - Reinhard Zumkeller, Oct 24 2010

Examples

			The triangle starts in row n=0 with columns 0<=m<=n as:
  1;
  1,  1;
  1,  0,  1;
  1,  3,  3,  1;
  1,  0,  0,  0,  1;
  1,  5,  0,  0,  5,  1;
  1,  0, 15,  0, 15,  0,  1;
  1,  7, 21, 35, 35, 21,  7,  1;
  1,  0,  0,  0,  0,  0,  0,  0,  1;
  1,  9,  0,  0,  0,  0,  0,  0,  9,  1;
  1,  0, 45,  0,  0,  0,  0,  0, 45,  0,  1;

Links

Crossrefs

Cf. A007318, A014421, A047999, A088560. - Reinhard Zumkeller, Oct 24 2010

Programs

  • Haskell
    a143333 n k = a143333_tabl !! (n-1) !! (k-1)
a143333_row n = a143333_tabl !! (n-1)
a143333_tabl = zipWith(zipWith (*)) a007318_tabl a047999_tabl
-- Reinhard Zumkeller, Oct 10 2013
  • Mathematica
    t[n_, m_] = Mod[Binomial[n, m], 2]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Formula

T(n,m) = A047999(n,m)*A007318(n,m).

Extensions

Offset set to 0 by Reinhard Zumkeller, Oct 21 2010

A014428 Even elements in Pascal's triangle.

Original entry on oeis.org

2, 4, 6, 4, 10, 10, 6, 20, 6, 8, 28, 56, 70, 56, 28, 8, 36, 84, 126, 126, 84, 36, 10, 120, 210, 252, 210, 120, 10, 330, 462, 462, 330, 12, 66, 220, 792, 924, 792, 220, 66, 12, 78, 286, 1716, 1716, 286, 78, 14, 364, 2002, 3432, 2002, 364, 14, 16, 120, 560, 1820
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Triangle begins:
2
4 6 4
10 10

Links

Crossrefs

Cf. A007318, A014421, A048967.

Programs

  • Mathematica
    Select[ Flatten[ Table[ Binomial[ n, i ], {n, 0, 20}, {i, 0, n} ] ], EvenQ[ # ]& ]

Extensions

More terms from Erich Friedman
Offset corrected by Mohammad K. Azarian, Nov 19 2008

A014414 Odd elements in Pascal's triangle that are not 1.

Original entry on oeis.org

3, 3, 5, 5, 15, 15, 7, 21, 35, 35, 21, 7, 9, 9, 45, 45, 11, 55, 165, 165, 55, 11, 495, 495, 13, 715, 1287, 1287, 715, 13, 91, 1001, 3003, 3003, 1001, 91, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 17, 17, 153, 153, 19, 171
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Irregular array begins:
  3, 3
  5, 5
  15, 15
  7, 21, 35, 35, 21, 7
  9, 9
  45, 45
  ...

Crossrefs

Cf. A007318, A014421.

Extensions

More terms from Erich Friedman
Offset corrected by Mohammad K. Azarian, Nov 19 2008

A348649 Odd numbers in the triangle of Stirling numbers of the second kind (A008277).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 1, 1, 15, 25, 1, 1, 31, 65, 15, 1, 1, 63, 301, 21, 1, 1, 127, 1701, 1, 1, 255, 3025, 6951, 1, 1, 511, 34105, 42525, 22827, 45, 1, 1, 1023, 28501, 179487, 63987, 1155, 55, 1, 1, 2047, 611501, 159027, 22275, 1705, 1, 1, 4095, 261625, 7508501, 39325, 2431, 1
Offset: 1

Views

Author

Rémy Sigrist, Oct 27 2021

Keywords

Comments

We take the odd values in A008277, as they appear, with duplicates.
For any n >= 1, the n-th row has A007306(n) terms.

Examples

			As an irregular table, the first rows are:
     1:    1;
     2:    1, 1;
     3:    1, 3, 1;
     4:    1, 7, 1;
     5:    1, 15, 25, 1;
     6:    1, 31, 65, 15, 1;
     7:    1, 63, 301, 21, 1;
     8:    1, 127, 1701, 1;
     9:    1, 255, 3025, 6951, 1;
    10:    1, 511, 34105, 42525, 22827, 45, 1;
    11:    1, 1023, 28501, 179487, 63987, 1155, 55, 1;
    ...

Links

Crossrefs

See A014421, A014428, A014450, A014459 for similar sequences.
Cf. A007306, A008277, A348650 (even numbers).

Programs

  • PARI
    row(n) = select(v -> v%2==1, vector(n, k, stirling(n, k, 2)))

A355604 Table T(n, k), n >= 0, k = 0..n, read by rows; row n is obtained by replacing in row n of Pascal's triangle (A007318) runs of k consecutive even numbers by the terms of row k+1 of the present triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 15, 1, 15, 1, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 1, 1, 15, 1, 15, 1, 1, 1, 1, 9, 1, 5, 1, 1, 5, 1, 9, 1, 1, 1, 45, 1, 1, 1, 1, 1, 45, 1, 1, 1, 11, 55, 165, 1, 3, 3, 1, 165, 55, 11, 1, 1, 1, 1, 1, 495, 1, 1, 1, 495, 1, 1, 1, 1
Offset: 0

Views

Author

Rémy Sigrist, Jul 09 2022

Keywords

Comments

This triangle has fractal features: even terms of Pascal's triangle are clustered as wXwXw subtriangles; these subtriangles are replaced by the first w rows (flipped upside-down) of the present triangle.

Examples

			Triangle T(n, k) begins (stars indicate replacements):
  n\k|   0    1    2    3    4    5    6    7    8    9   10   11   12
  ---+-----------------------------------------------------------------
    0|   1
    1|   1    1
    2|   1    1*   1
    3|   1    3    3    1
    4|   1    1*   1*   1*   1
    5|   1    5    1*   1*   5    1
    6|   1    1*  15    1*  15    1*   1
    7|   1    7   21   35   35   21    7    1
    8|   1    1*   1*  15*   1*  15*   1*   1*   1
    9|   1    9    1*   5*   1*   1*   5*   1*   9    1
   10|   1    1*  45    1*   1*   1*   1*   1*  45    1*   1
   11|   1   11   55  165    1*   3*   3*   1* 165   55   11    1
   12|   1    1*   1*   1* 495    1*   1*   1* 495    1*   1*   1*   1

Links

Crossrefs

Cf. A007318, A065040, A014421, A143333, A348648.

Programs

  • PARI
    row(n) = { my (r=binomial(n)); for (i=1, #r, if (r[i]%2==0, for (w=1, oo, if (r[i+w]%2==1, my (t=row(w-1)); for (j=1, #t, r[i-1+j]=t[j]); i+=w; break)))); return (r) }

A014726 Squares of odd elements in Pascal triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 9, 9, 1, 1, 1, 1, 25, 25, 1, 1, 225, 225, 1, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 1, 1, 81, 81, 1, 1, 2025, 2025, 1, 1, 121, 3025, 27225, 27225, 3025, 121, 1, 1, 245025, 245025, 1, 1, 169, 511225, 1656369, 1656369
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Crossrefs

Cf. A014421.

Formula

a(n) = A014421(n)^2. - Sean A. Irvine, Nov 18 2018

Extensions

More terms from Sean A. Irvine, Nov 18 2018
Showing 1-6 of 6 results.

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