

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
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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 basep digit of i exceeds the corresponding basep digit of j.
If p is prime, T(n*p,p) = (p*(p+1)/2)*T(n,p) + p*(p1)*n*(n1)/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..n1)], 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, n1, 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



