A378006 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.

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, 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, 0, 0, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Jianing Song, Nov 14 2024

Keywords

• nonn
• tabl
• easy

Comments

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).

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)}.

Examples

			Table starts
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 2, 0, 0, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 2, 0, 0, 2, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, 0, 2, 0, 0, 0, ...
  1, 1, 0, 1, 0, 0, 0, 2, 0, 0, ...
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
See A378007 for more details.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..11325 (the first 150 diagonals, with n+k = 2..151)
• J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973.

Crossrefs

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

Cf. A378007.

Programs

PARI

A378006(n,k) = {
my(f = factor(n), res = 1); for(i=1, #f~, if(k % f[i,1] == 0, return(0));
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)));
res;}

Formula

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.

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