A014428 -id:A014428 - cp's OEIS frontend

A348648 Triangle T(n, k), n >= 0, 0 <= k <= n, read by rows; the even terms of Pascal's triangle (A007318) are clustered as wXwXw subtriangles with w = 2^s-1 for some s > 0; if A007318(n, k) is even then T(n, k) gives the corresponding s otherwise T(n, k) = 0.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 3, 3, 3, 3, 0, 0, 0, 3, 3, 3, 3, 3, 3, 0, 0, 0, 1, 0, 3, 3, 3, 3, 3, 0, 1, 0, 0, 0, 0, 0, 3, 3, 3, 3, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 3, 3, 0, 2, 2, 2, 0

Offset: 0

Rémy Sigrist, Oct 27 2021

It is possible to build this triangle with the following procedure:
- T_0 is the 2X2X2 triangle with all 0's:
          ^
  T_0 =  /0\
        /0 0\
       +-----+
- for s > 0, T_s is obtained by arranging 3 copies of T_{s-1} and one wXwXw triangle (where w=2^s-1) will all s's as follows:
                 ^
                / \
               /   \
              /     \
             /T_{s-1}\
  T_s =     +---------+
           / \s ... s/ \
          /   \.   ./   \
         /     \. ./     \
        /T_{s-1}\s/T_{s-1}\
       +---------+---------+
- the triangle {T(n, k)} is the limit of T_s as s tends to infinity.
For any n >= 0, A001316(n) gives the number of 1's in row 4*n + 2 (and there are no 1's elsewhere).

			Triangle T(n, k) begins (with dots instead of 0's):
                    .
                   . .
                  . 1 .
                 . . . .
                . 2 2 2 .
               . . 2 2 . .
              . 1 . 2 . 1 .
             . . . . . . . .
            . 3 3 3 3 3 3 3 .
           . . 3 3 3 3 3 3 . .
          . 1 . 3 3 3 3 3 . 1 .
         . . . . 3 3 3 3 . . . .
        . 2 2 2 . 3 3 3 . 2 2 2 .
       . . 2 2 . . 3 3 . . 2 2 . .
      . 1 . 2 . 1 . 3 . 1 . 2 . 1 .
     . . . . . . . . . . . . . . . .

Rémy Sigrist, Table of n, a(n) for n = 0..8255 (rows for n = 0..127, flattened)
Rémy Sigrist, Colored representation of the first 2^9 rows (black pixels correspond to 0's)
Rémy Sigrist, Colored logarithmic scatterplot of A014428 (where the colors are given by the present sequence)
Rémy Sigrist, Colored logarithmic scatterplot of A007318 (where the colors are given by the present sequence, and black pixels denotes 0's)
Index entries for triangles and arrays related to Pascal's triangle

Cf. A001316, A007318, A047999, A014428, A129760.

T(n,k) = { if (n<=1, return (0)); my (s=#binary(n)-1); n-=2^s; if (k<=n, return (T(n,k)), k<2^s, return (s), return (T(n,k-2^s))) }

T(n, k) = 0 iff A047999(n, k) = 1.

2^T(n, 0) OR 2^T(n, 1) OR ... OR 2^T(n, n) = 1 + A129760(n+1) (where OR denotes the bitwise OR operator).

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

Rémy Sigrist, Oct 27 2021

We take the odd values in A008277, as they appear, with duplicates.

For any n >= 1, the n-th row has A007306(n) terms.

			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;
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..10057 (rows for n = 1..331, flattened)
Rémy Sigrist, Logarithmic scatterplot of the first 35000 terms
Index entries for sequences related to Stirling numbers

See A014421, A014428, A014450, A014459 for similar sequences.

Cf. A007306, A008277, A348650 (even numbers).

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

A014727 Squares of even elements in Pascal's triangle A007318.

Original entry on oeis.org

4, 16, 36, 16, 100, 100, 36, 400, 36, 64, 784, 3136, 4900, 3136, 784, 64, 1296, 7056, 15876, 15876, 7056, 1296, 100, 14400, 44100, 63504, 44100, 14400, 100, 108900, 213444, 213444, 108900, 144, 4356, 48400, 627264, 853776, 627264, 48400, 4356

Offset: 0

Mohammad K. Azarian

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A007318, A014428.

Select[Flatten[Table[Binomial[n,k],{n,0,20},{k,0,n}]],EvenQ]^2 (* Harvey P. Dale, Apr 18 2020 *)

a(n) = A014428(n)^2. - Sean A. Irvine, Nov 18 2018

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

cp's OEIS Frontend

A348648 Triangle T(n, k), n >= 0, 0 <= k <= n, read by rows; the even terms of Pascal's triangle (A007318) are clustered as wXwXw subtriangles with w = 2^s-1 for some s > 0; if A007318(n, k) is even then T(n, k) gives the corresponding s otherwise T(n, k) = 0.

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

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Author

Keywords

Comments

Examples

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Programs

PARI

A014727 Squares of even elements in Pascal's triangle A007318.

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Mathematica

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