A378006 - OEIS

%I #15 Nov 14 2024 23:23:53

%S 1,1,1,1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0,1,1,0,0,0,2,0,

%T 1,1,1,0,0,0,0,0,2,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,

%U 0,0,0,0,2,0,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1

%N Square table read by descending antidiagonals: the k-th column has Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k.

%C For fixed k, we have Product_{chi} L(chi,s) = Product_{p not dividing k} 1/(1 - 1/p^(ord(p,k)*s))^(phi(k)/ord(p,k)), where phi = A000010 is the Euler totient function and ord(a,k) is the multiplicative order of a modulo k; see Section 3.4 of Chapter VI, Proposition 13, page 72 of J.-P. Serre, A Course in Arithmetic. Using the series expansion of 1/(1-x)^r, we get Product_{chi} L(chi,s) = Product_{p not dividing k} (Sum_{n>=0} binomial(n+phi(k)/ord(p,k)-1,phi(k)/ord(p,k)-1)/p^(ord(p,k)*s)), giving us the formula to calculate T(n,k).

%C From the formula we can wee that T(n,k) = 0 unless n == 1 (mod k). A378007 is the condensed version giving only {T(k*n+1,k)}.

%H Jianing Song, <a href="/A378006/b378006.txt">Table of n, a(n) for n = 1..11325</a> (the first 150 diagonals, with n+k = 2..151)

%H J.-P. Serre, <a href="https://www.math.purdue.edu/~jlipman/MA598/Serre-Course%20in%20Arithmetic.pdf">A Course in Arithmetic</a>, Springer-Verlag, 1973.

%F Each column is multiplicative: T(p^e,k) = 0 if p divides k; 0 if e is not divisible by ord(p,k); binomial(e/ord(p,k)+phi(k)/ord(p,k)-1,phi(k)/ord(p,k)-1) otherwise.

%F For odd k, T(2*k,n) = T(k,n) for odd n, 0 for even n.

%e Table starts

%e   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e   1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e   1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...

%e   1, 1, 0, 2, 0, 0, 0, 0, 0, 0, ...

%e   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e   1, 1, 2, 0, 0, 2, 0, 0, 0, 0, ...

%e   1, 0, 0, 0, 0, 0, 2, 0, 0, 0, ...

%e   1, 1, 0, 1, 0, 0, 0, 2, 0, 0, ...

%e   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e See A378007 for more details.

%o (PARI) A378006(n,k) = {

%o my(f = factor(n), res = 1); for(i=1, #f~, if(k % f[i,1] == 0, return(0));

%o my(d = znorder(Mod(f[i,1],k))); if(f[i,2] % d != 0, return(0), my(m = f[i,2]/d, r = eulerphi(k)/d); res *= binomial(m+r-1,r-1)));

%o res;}

%Y Columns: A000012 (k=1), A000035 (k=2), A045833 (k=3), A008442 (k=4).

%Y Cf. A378007.

%K nonn,tabl,easy

%O 1,33

%A _Jianing Song_, Nov 14 2024