A378007 Square table read by descending antidiagonals: T(n,k) = A378006(k*n+1,k).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 4, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 0, 2, 0, 0, 2, 1, 1, 1, 0, 4, 0, 1, 0, 3, 0, 1, 1, 1, 4, 6, 2, 6, 2, 4, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 1, 1, 4, 10, 4, 6, 4, 6, 2, 4, 2, 2, 1, 1

Offset

0,12

Comments

A condensed version of A378006: the k-th column is the sequence {b(k*n+1)}, with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k.

Links

Jianing Song, Table of n, a(n) for n = 0..11324 (the first 150 diagonals, with n+k = 1..150)

Formula

See A378006.

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

Example

Table starts
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 1, 2, 0, 2, 2, 2, 0, 4, ...
  1, 1, 2, 1, 4, 2, 0, 4, 6, 0, ...
  1, 1, 0, 2, 1, 2, 0, 2, 0, 4, ...
  1, 1, 2, 2, 0, 1, 6, 0, 6, 4, ...
  1, 1, 1, 0, 0, 2, 0, 4, 0, 0, ...
  1, 1, 2, 3, 4, 2, 6, 2, 0, 4, ...
  1, 1, 0, 2, 0, 2, 0, 0, 1, 4, ...
  1, 1, 1, 0, 4, 3, 0, 0, 6, 1, ...
  1, 1, 2, 2, 0, 0, 3, 4, 0, 0, ...
  1, 1, 2, 2, 0, 2, 6, 3, 0, 4, ...
Write w = exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2.
Column k = 1: 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + 1/7^s + 1/8^s + 1/9^s + 1/10^s + 1/11^s + ...;
Column k = 2: 1 + 1/3^s + 1/5^s + 1/7^s + 1/9^s + 1/11^s + 1/13^s + 1/15^s + 1/17^s + 1/19^s + 1/21^s + ...;
Column k = 3: (1 + 1/2^s + 1/4^s + 1/5^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + ...) = 1 + 1/4^s + 2/7^s + 2/13^s + 1/16^s + 2/19^s + 1/25^s + 2/28^s + 2/31^s + ...;
Column k = 4: (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...) = 1 + 2/5^s + 1/9^s + 2/13^s + 2/17^s + 3/25^s + 2/29^s + 2/37^s + 2/41^s + ...;
Column k = 5: (1 + 1/2^s + 1/3^s + 1/4^s + ...)*(1 + i/2^s - i/3^s - 1/4^s + ...)*(1 - 1/2^s - 1/3^s + 1/4^s + ...)*(1 - i/2^s + i/3^s - 1/4^s + ...) = 1 + 4/11^s + 1/16^s + 4/31^s + 4/41^s + ...;
Column k = 6: (1 + 1/5^s + 1/7^s + 1/11^s + ...)*(1 - 1/5^s + 1/7^s - 1/11^s + ...) = 1 + 2/7^s + 2/13^s + 2/19^s + 1/25^s + 1/31^s + 2/37^s + 2/43^s + 3/49^s + 2/61^s + ...;
Column k = 7: (1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + ...)*(1 + w/2^s + (w+1)/3^s - (w+1)/4^s - w/5^s - 1/6^s + ...)*(1 - (w+1)/2^s + w/3^s + w/4^s - (w+1)/5^s + 1/6^s + ...)*(1 + 1/2^s - 1/3^s + 1/4^s - 1/5^s - 1/6^s + ...)*(1 + w/2^s - (w+1)/3^s - (w+1)/4^s + w/5^s + 1/6^s + ...)*(1 - (w+1)/2^s - w/3^s + w/4^s + (w+1)/5^s - 1/6^s + ...) = 1 + 2/8^s + 6/29^s + 6/43^s + 3/64^s + 6/71^s + ...;
Column k = 8: (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 + 1/3^s - 1/5^s - 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...)*(1 - 1/3^s - 1/5^s + 1/7^s + ...) = 1 + 2/9^s + 4/17^s + 2/25^s + 4/41^s + 2/49^s + 4/73^s + 3/81^s + ...;
Column k = 9: (1 + 1/2^s + 1/4^s + 1/5^s + 1/7^s + 1/8^s + ...)*(1 + (w+1)/2^s + w/4^s - w/5^s - (w+1)/7^s - 1/8^s + ...)*(1 + w/2^s - (w+1)/4^s - (w+1)/5^s + w/7^s + 1/8^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + 1/7^s - 1/8^s + ...)*(1 - (w+1)/2^s + w/4^s + w/5^s - (w+1)/7^s + 1/8^s + ...)*(1 - w/2^s - (w+1)/4^s + (w+1)/5^s + w/7^s - 1/8^s + ...) = 1 + 6/19^s + 6/37^s + 1/64^s + 6/73^s + ...;
Column k = 10: (1 + 1/3^s + 1/7^s + 1/9^s + ...)*(1 + i/3^s - i/7^s - 1/9^s + ...)*(1 - 1/3^s - 1/7^s + 1/9^s + ...)*(1 - i/3^s + i/7^s - 1/9^s + ...) = 1 + 4/11^s + 4/31^s + 4/41^s + 4/61^s + 4/71^s + 1/81^s + 4/101^s + ...

Prog

(PARI) A378007(n, k) = {
my(f = factor(k*n+1), res = 1); for(i=1, #f~, 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 and k=2), A033687 (k=3), A008441 (k=4), A378008 (k=5), A097195 (k=6), A378009 (k=7), A378010 (k=8), A378011 (k=9), A378012 (k=10).

Cf. A378006.

Sequence in context: A004578 A319372 A319367 * A234514 A343453 A051031

Adjacent sequences: A378004 A378005 A378006 * A378008 A378009 A378010

Keyword

nonn,tabl,easy

Author

Jianing Song, Nov 14 2024

Status

approved