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A267959 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. 1

%I

%S 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,

%T 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,

%U 1,0,0,0,0,0,0,1,1,1,1,1,0,1,1,1,0,1,1,1,0,1

%N 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.

%C 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)).

%C 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).

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

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

%H Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

%H Tom Edgar, <a href="/A267959/a267959.pdf">Triangular array image.</a> 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.

%o (Sage)

%o def carry_sequence(n,k,p):

%o M=(n-k).digits(base=p)

%o K=k.digits(base=p)

%o mm=max(len(K),len(M))

%o M=M+(mm-len(M)+1)*[0]

%o K=K+(mm-len(K)+1)*[0]

%o CS=[floor((M[0]+K[0])/p)]

%o for i in [1..mm]:

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

%o return CS

%o def checkcarrysequence(n,k,p):

%o CS=carry_sequence(n,k,p)

%o if 0 in CS:

%o T=CS[CS.index(0):]

%o if T==len(T)*[0]:

%o return true

%o else:

%o return false

%o else:

%o return true

%o def T(n,k):

%o flag=true

%o for x in prime_range(n+1):

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

%o flag=false

%o return Integer(flag)

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

%o [x for sublist in T for x in sublist]

%K nonn,tabl

%O 0

%A _Tom Edgar_ and Michael Z. Spivey, Jan 22 2016

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Last modified October 17 13:10 EDT 2021. Contains 348048 sequences. (Running on oeis4.)