A268978 Triangle T(n,k) read by rows with 1 <= k <= n: number of entries in the first n rows of Pascal's triangle that are divisible by k.

1, 3, 0, 6, 1, 0, 10, 1, 2, 0, 15, 4, 3, 2, 0, 21, 6, 3, 2, 4, 1, 28, 9, 7, 3, 7, 3, 0, 36, 9, 9, 3, 9, 3, 6, 0, 45, 16, 9, 9, 10, 3, 11, 4, 0, 55, 22, 17, 13, 10, 9, 15, 4, 6, 4, 66, 29, 24, 16, 18, 14, 18, 6, 9, 10, 0, 78, 33, 30, 16, 24, 18, 20, 6, 9, 12

Offset

1,2

Comments

T(n, 1) = A000217(n).

T(n, 2) = A051679(n - 1) for n > 0.

From Robert Israel, Feb 26 2016: (Start)

If p is prime, T(n,p) is the number of pairs (i,j) with 0 < i < j < n such that some base-p digit of i exceeds the corresponding base-p digit of j.

If p is prime, T(n*p,p) = (p*(p+1)/2)*T(n,p) + p*(p-1)*n*(n-1)/4. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Example

Triangle begins:
   1
   3 0
   6 1 0
  10 1 2 0
  15 4 3 2 0
  21 6 3 2 4 1
  28 9 7 3 7 3 0
  36 9 9 3 9 3 6 0
T(5, 2) = 4 because the first five rows of Pascal's triangle have 4 entries divisible by 2: one entry in the third row, and three entries in the fifth row.
      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1

Maple

T:= (n, k) ->  numboccur([seq(seq(binomial(m, j) mod k, j=0..m), m=0..n-1)], 0):
seq(seq(T(n, k), k=1..n), n=1..10); # Robert Israel, Feb 26 2016

Prog

(PARI) T(n, k) = sum(m=0, n-1, sum(j=0, m, (binomial(m, j) % k) == 0)); \\ Michel Marcus, Feb 23 2016

Crossrefs

Cf. A000217, A051679.

Sequence in context: A081978 A117784 A257896 * A280585 A359218 A197807

Adjacent sequences: A268975 A268976 A268977 * A268979 A268980 A268981

Keyword

nonn,tabl

Author

Peter Kagey, Feb 16 2016

Status

approved