The number of binomial coefficients divisible by a fixed power of 2

Abstract

Abstract.Define f(n,j) as the number of binomial coefficients (?) divisible by exactly 2». We find formulas for/(n, j) for 0â_j'ë4 and evaluate/(n, j) for special values of n.

Divisibility of binomial coefficients by powers of two

“…, p − 1}, and each variable X w is set to |n| w . For example, we have the following formulas, found by Howard [20]: The number of terms in these expressions is sequence A275012 in Sloane's OEIS [29] and can be seen as a measure of complexity of the sequence n → ϑ 2 (j, n). This was noted by Rowland (see the comments to A001316, A163000 and A163577 in the OEIS).…”

Subprime Factorization and the Numbers of Binomial Coefficients Exactly Divided by Powers of a Prime

“…Howard gave formulas for  j .n; 2/ and for  2 .n; p/ in 1971 and 1973 ( [5,6]). We recently gave the following general formula for  j .n; p/ ( [2]).…”

“…It is now easy to prove general recurrence relation (5). Clearly, for r > 0 and c r > 1, ‚.n; p/ D ‚.n .c r 1/p r ; p/j A p r ;2p 1 1 C c r ‚.n .c r 1/p r ; p/j R p r ;2p 1 1 D .c r C 1/‚.n .c r 1/p r ; p/j A p r ;2p 1 1 C c r ‚.n .c r 1/p r ; p/j V p r ;2p 1 1 :…”

“…Substituting the expressions with unrestricted ‚ in (8) and combining "like" terms, we obtain general recurrence relation (5).…”

Distribution of Binomial Coefficients and Digital Functions

“…It is the main combinatorial tool for the study of the number of binomial coefficients in a given row of Pascal's triangle divisible by a given power of a prime. Partial results and exact formulae in relation to this were obtained in [6,25,26].…”