A051679 -id:A051679 - cp's OEIS frontend

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

Peter Kagey, Feb 16 2016

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)

			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

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

Cf. A000217, A051679.

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

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

cp's OEIS Frontend

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.

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