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A065040 Triangle read by rows: T(m,k) = exponent of the highest power of 2 dividing the binomial coefficient binomial(m,k). 5
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 3, 1, 3, 2, 3, 0, 0, 0, 2, 2, 1, 1, 2, 2, 0, 0, 0, 1, 0, 3, 1, 2, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

T(m,k) is the number of 'carries' that occur when adding k and m-k in base 2 using the traditional addition algorithm. - Tom Edgar, Jun 10 2014

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10010; the first 141 antidiagonals, flattened

Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

FORMULA

As an array f(i,j) = f(j,i) = T(i+j,j) read by antidiagonals: f(0,j) = 0, f(1,j) = A007814(j+1), f(i,j) = Sum_{k=0..i-1} (f(1,j+k) - f(1,k)). [corrected by Kevin Ryde, Oct 07 2021]

The n-th term a(n) is equal to the binomial coefficient binomial(m,k), where m = floor((1+sqrt(8*n+1))/2) - 1 and k = n - m(m+1)/2. Also a(n) = g(m) - g(k) - g(m-k), where g(x) = Sum_{i=1..floor(log_2(x))} floor(x/2^i), m = floor((1+sqrt(8*n+1))/2) - 1, k = n - m(m+1)/2. - Hieronymus Fischer, May 05 2007

T(m,k) <= log_2 m, for m > 0. - Charles R Greathouse IV, Mar 26 2013

T(m,k) = log_2(A082907(m,k)). - Tom Edgar, Jun 10 2014

From Antti Karttunen, Oct 28 2014: (Start)

a(n) = A007814(A007318(n)).

a(n) * A047999(n) = 0 and a(n) + A047999(n) > 0 for all n.

(End)

EXAMPLE

Triangle begins:

[0]

[0, 0]

[0, 1, 0]

[0, 0, 0, 0]

[0, 2, 1, 2, 0]

[0, 0, 1, 1, 0, 0]

[0, 1, 0, 2, 0, 1, 0]

[0, 0, 0, 0, 0, 0, 0, 0]

[0, 3, 2, 3, 1, 3, 2, 3, 0]

[0, 0, 2, 2, 1, 1, 2, 2, 0, 0]

[0, 1, 0, 3, 1, 2, 1, 3, 0, 1, 0]

... - N. J. A. Sloane, Aug 21 2021

MAPLE

A065040 := (n, k) -> padic[ordp](binomial(n, k), 2):

seq(seq(A065040(n, k), k=0..n), n=0..13); # Peter Luschny, Aug 15 2017

MATHEMATICA

T[m_, k_] := IntegerExponent[Binomial[m, k], 2]; Table[T[m, k], {m, 0, 13}, {k, 0, m}] // Flatten (* Jean-François Alcover, Oct 06 2016 *)

PROG

(PARI) T(m, k)=hammingweight(k)+hammingweight(m-k)-hammingweight(m)

for(m=0, 9, for(k=0, m, print1(T(m, k)", "))) \\ Charles R Greathouse IV, Mar 26 2013

CROSSREFS

Row sums: A187059.

Cf. A007318, A007814, A001511, A000120, A047999, A049606, A000680, A048881, A011371, A005187, A000265, A001316, A001317, A243759.

Sequence in context: A079071 A322795 A050602 * A284688 A057595 A035201

Adjacent sequences:  A065037 A065038 A065039 * A065041 A065042 A065043

KEYWORD

nonn,tabl,easy,changed

AUTHOR

Claude Lenormand (hlne.lenormand(AT)voono.net), Nov 05 2001

EXTENSIONS

Name clarified by Antti Karttunen, Oct 28 2014

STATUS

approved

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Last modified October 18 08:58 EDT 2021. Contains 348067 sequences. (Running on oeis4.)