We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$.
arXiv:0710.1468v1 [math.GM] 8 Oct 2007
Number of binomial coefficients divided by a
fixed power of a prime
William B. Everett
Chernogolovka, Moscow Oblast, Russia
November 4, 2018
Abstract
We state a general formula for the number of binomial coefficients n
choose k that are divided by a fixed power of a prime p, i.e., the number
of binomial coefficients divided by pj and not divided by pj+1.
Let n be a natural number and p be a prime. Let θj(n) denote the number
of binomial coefficients nCk = ( n
k ) such that pj divides nCk and pj+1 does not
divide nCk.
We represent n in the base p: n = c0 + c1p + c2p2 + · · · + crpr, 0 ≤ci < p,
i = 0, 1, . . . , r, cr ̸= 0.
Let W be the set of r-bit binary words, i.e.,
W =
w = w1w2 . . . wr : wi ∈{0, 1}, 1 ≤i ≤r
.
We partition W into r+1 subsets Wj, 0 ≤j ≤r, where
Wj =
w ∈W :
r
X
i=1
wi = j
.
We define the functions F(w), L(w), and M(w, i) as follows:
F(w) =
(
c0 + 1
if w1 = 0,
p −c0 −1
if w1 = 1,
L(w) =
(
cr + 1
if wr = 0,
cr
if wr = 1,
M(w, i) =
ci + 1
if wi = 0 and wi+1 = 0,
p −ci −1
if wi = 0 and wi+1 = 1,
ci
if wi = 1 and wi+1 = 0,
p −ci
if wi = 1 and wi+1 = 1.
1
The general formula for θj(n) is
θj(n) =
X
w∈Wj
F(w)L(w)
r−1
Y
i=1
M(w, i).
(1)
Obviously, we have the sum of rCj terms and each term is the product of r+1
factors. It is easy to establish that if p−cℓ−1=0 for some ℓ, then p−ci −1 = 0
for all i ≤ℓ(and some terms may vanish from the sum). It is also easy to
establish that if p −cℓ−1 = 1 for some ℓ, then p −ci −1 = 1 for all i ≤ℓ(and
the number of contributing factors in some terms is reduced). This means that
the formula can be simplified for n of certain special forms.
Formula (1) reproduces known formulas for some particular values.
For
example, for j = 0, we obtain the known formula [1]
θ0(n) = (c0 + 1)(cr + 1)(c1 + 1) · · · (cr−1 + 1)
= (c0 + 1)(c1 + 1) · · · (cr + 1).
For j = 1, we obtain the known formula [2]
θ1(n) = (c0 + 1)cr(c1 + 1)(c2 + 1) · · · (cr−2 + 1)(p −cr−1 −1)
- (c0 + 1)(cr + 1)(c1 + 1)(c2 + 1) · · · (p −cr−2 −1)cr−1
- · · ·
- (c0 + 1)(cr + 1)(p −c1 −1)c2(c3 + 1) · · · (cr−1 + 1)
- (p −c0 −1)(cr + 1)c1(c2 + 1) · · · (cr−1 + 1)
=
r−1
X
k=0
(c0 + 1) · · · (ck−1 + 1)(p −ck −1)ck+1(c + k + 2 + 1) · · · (cr + 1).
Other particular formulas for θj(n) can be found in [3] and [4].
References
[1] Fine, N. J.: Binomial coefficients modulo a prime. Amer. Math. Monthly
54, 589–592 (1947)
[2] Carlitz, L.: The number of binomial coefficients divisible by a fixed power
of a prime. Rend. Circ. Mat. Palermo (2) 16, 299–320 (1967)
[3] Howard, F. T.: The number of binomial coefficients divisible by a fixed
power of 2. Proc. Amer. Math. Soc. 29, 236–242 (1971)
[4] Howard, F. T.: Formulas for the number of binomial coefficients divisible by
a fixed power of a prime. Proc. Amer. Math. Soc. 37, 358–362 (1973)
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