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

Author

Jianing Song, Nov 14 2024

Keywords

• nonn
• tabl
• easy

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.

Examples

			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 + ...
		

Links

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

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.

Programs

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;}

Formula

See A378006.

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