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A267959 Triangle read by rows: T(n,k) = 1 if the generalized binomial coefficient (n,k)_f is integral for every multiplicative function f; otherwise T(n,k) = 0. 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0

COMMENTS

For 0<=k<=n, we define (n,k)_f := Product_{i=1..n}f(i)/(Product_{i=1..k}f(i) * Product_{i=1..n-k}f(i)).

T(n,k) = 1 if and only if for every prime p<=n there exists an index s_p>=0 such that e(n,n-k,i,p) = 1 for all 0<= i < s_p and e(n,n-k,i,p) = 0 for all i >= s_p where e(n,n-k,i,p) represents the value of the carry in the i-th position when adding the base-p representations of n and n-k (See Corollary 12 in Edgar-Spivey reference).

T(n,0) = 1 and T(n,1) = 1 for all n.

T(n,2) = 1 if and only if n == 2 (mod 4) or n == 3 (mod 4).

LINKS

Table of n, a(n) for n=0..89.

Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Tom Edgar, Triangular array image. This image is Figure 1 in Edgar-Spivey reference; it shows rows 0-90 of the triangle with shaded entries corresponding to 1 and other entries corresponding to 0.

PROG

(Sage)

def carry_sequence(n, k, p):

    M=(n-k).digits(base=p)

    K=k.digits(base=p)

    mm=max(len(K), len(M))

    M=M+(mm-len(M)+1)*[0]

    K=K+(mm-len(K)+1)*[0]

    CS=[floor((M[0]+K[0])/p)]

    for i in [1..mm]:

        CS.append(floor((M[i]+K[i]+CS[i-1])/p))

    return CS

def checkcarrysequence(n, k, p):

    CS=carry_sequence(n, k, p)

    if 0 in CS:

        T=CS[CS.index(0):]

        if T==len(T)*[0]:

            return true

        else:

            return false

    else:

        return true

def T(n, k):

    flag=true

    for x in prime_range(n+1):

        if not(checkcarrysequence(n, k, x)):

            flag=false

    return Integer(flag)

T=[[T(i, j) for j in [0..i]] for i in [0..20]]

[x for sublist in T for x in sublist]

CROSSREFS

Sequence in context: A105567 A114213 A108358 * A144384 A144475 A011758

Adjacent sequences:  A267956 A267957 A267958 * A267960 A267961 A267962

KEYWORD

nonn,tabl

AUTHOR

Tom Edgar and Michael Z. Spivey, Jan 22 2016

STATUS

approved

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Last modified December 4 09:05 EST 2020. Contains 338921 sequences. (Running on oeis4.)