A220645 T(n,k): number of binomial coefficients C(n,r), for 0 <= r <= n, divisible by 2^k.

1, 2, 0, 3, 1, 0, 4, 0, 0, 0, 5, 3, 2, 0, 0, 6, 2, 0, 0, 0, 0, 7, 3, 1, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 9, 7, 6, 4, 0, 0, 0, 0, 0, 10, 6, 4, 0, 0, 0, 0, 0, 0, 0, 11, 7, 3, 2, 0, 0, 0, 0, 0, 0, 0, 12, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 9, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Michel Marcus, Dec 17 2012

Keywords

• nonn
• tabl

Comments

Since the sum of binomial coefficients of the n-th row of Pascal's triangle is 2^n, T(n, k)=0 for k >= n. So only n elements, from k=0 to n-1, will be displayed at row n, giving a triangle instead of a table.

The number A119387(n) gives the position of the last positive number in each row. - T. D. Noe, Dec 18 2012

Examples

			Triangle starts:
0: 1
1: 2 0
2: 3 1 0
3: 4 0 0 0
4: 5 3 2 0 0
5: 6 2 0 0 0 0
For n=4, the corresponding Pascal's triangle row is:
1 4 6 4 1,
with 5 numbers divisible by 2^0,
and 3 numbers divisible by 2^1,
and 2 numbers divisible by 2^2,
and 0 numbers divisible by 2^3,
and 0 numbers divisible by 2^4.
		

Links

• T. D. Noe, Rows n = 0..100 of triangle, flattened
• F. T. Howard, The number of binomial coefficients divisible by a fixed power of 2, Proc. Amer. Math. Soc. 29 (1971), 236-242

Crossrefs

Cf. A000079, A007318.

Programs

Mathematica

Flatten[Table[b = Binomial[n, Range[0, n]]; Table[Count[b, ?(Mod[#, 2^k] == 0 &)], {k, 0, n}], {n, 0, 12}]] (* _T. D. Noe, Dec 18 2012 *)