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A267959
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Triangle read by rows: T(n,k) = 1 if the generalized binomial coefficient (n,k)_f is an integer for every multiplicative function f; otherwise T(n,k) = 0.
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1
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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
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OFFSET
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0
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COMMENTS
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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).
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LINKS
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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.
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PROG
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(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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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