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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
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 (list; table; graph; refs; listen; history; text; internal format)
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 A197807 A111074

Adjacent sequences:  A268975 A268976 A268977 * A268979 A268980 A268981

KEYWORD

nonn,tabl

AUTHOR

Peter Kagey, Feb 16 2016

STATUS

approved

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Last modified December 15 00:30 EST 2019. Contains 329988 sequences. (Running on oeis4.)