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.

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

Author

Rémy Sigrist, Jul 09 2022

Keywords

• nonn
• look
• tabl

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

• Rémy Sigrist, Table of n, a(n) for n = 0..8384 (rows for n = 0..128 flattened)
• Rémy Sigrist, Colored representation of the first 2^10 rows (where the hue is function of T(n, k), black pixels correspond to 1's)
• Index entries for triangles and arrays related to Pascal's triangle

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) }