A082907 A modified Pascal's triangle, read by rows, and modified as follows: binomial(n,j) is replaced by gcd(2^n, binomial(n,j)), i.e., the largest power of 2 dividing binomial(n,j).

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 2, 4, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 8, 2, 8, 4, 8, 1, 1, 1, 4, 4, 2, 2, 4, 4, 1, 1, 1, 2, 1, 8, 2, 4, 2, 8, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 4, 2, 4, 1, 8, 4, 8, 1, 4, 2, 4, 1, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1

Offset

0,5

Comments

If N is a power of 2, then the first N rows are invariant under all 6 symmetries of an equilateral triangle. - Paul Boddington, Dec 17 2003

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

E. Burlachenko, Fractal generalized Pascal matrices, arXiv:1612.00970 [math.NT], 2016. See p. 5.

Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Formula

From Paul Boddington, Dec 17 2003: (Start)

T(n, j) = c(n)/(c(j)*c(n-j)) where c(n)=A060818(n).

T(n, j) = (b(j)*b(n-j))/b(n) where b(n)=A001316(n) (Gould's sequence). (End)

Example

Triangle read by rows:
            1,
           1,1,
          1,2,1,
         1,1,1,1,
        1,4,2,4,1,
       1,1,2,2,1,1,
      1,2,1,4,1,2,1,
     1,1,1,1,1,1,1,1,
    1,8,4,8,2,8,4,8,1,
   1,1,4,4,2,2,4,4,1,1,
  ...
For n = -1 + 2^k, such rows consist of all 1's since all binomial coefficients C(n,j) are odd.

Mathematica

Flatten[Table[Table[GCD[2^n, Binomial[n, j]], {j, 0, n}], {n, 0, 25}], 1]
f[n_] := Denominator[CatalanNumber[n - 1]/2^(n - 1)]; T[n_, k_] := f[n]/(f[k]*f[n - k]); Table[T[n, k], {n, 0, 7}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 24 2016 *)

Crossrefs

Cf. A000005, A000079, A001316, A007318, A060818.

Sequence in context: A205399 A135303 A036065 * A146532 A305720 A225372

Adjacent sequences: A082904 A082905 A082906 * A082908 A082909 A082910

Keyword

nonn,tabl

Author

Labos Elemer, Apr 23 2003

Extensions

Edited by Jon E. Schoenfield, Dec 24 2016

Status

approved