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

1,33

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)}.

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.

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.

Example

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.

Prog

(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; }

Crossrefs

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

Cf. A378007.

Sequence in context: A320655 A359786 A359763 * A277017 A178498 A353422

Adjacent sequences: A378002 A378003 A378004 * A378007 A378008 A378009

Keyword

nonn,tabl,easy

Author

Jianing Song, Nov 14 2024

Status

approved