OFFSET
1,4
COMMENTS
LINKS
Seiichi Manyama, Rows n=1..140 of triangle, flattened
Gevorg Hmayakyan, On The Moebius and Euler Totient Functions Calculation.
Charles A. Nicol, On Restricted Partitions and a Generalization Of The Euler Totient and The Moebius Function, PNAS 39(9) (1953), 963-968.
FORMULA
a(n,t) = Sum(b(n, k*n + t), k=0..N(n, t)), where b(n,k) = A231599(n-1,k) and N(n,t) = [(n - 1)/2 - t/n].
a(n,t) = c_n(t) for t >= 1, where c_n(t) is a Ramanujan's sum A054533.
a(n,t) = a(n,-t)
From Seiichi Manyama, Mar 05 2018: (Start)
a(n,t) = c_n(n-t) = Sum_{d | gcd(n,n-t)} d*mu(n/d) for 0 <= t <= n-1.
So a(n,t) = Sum_{d | gcd(n,t)} d*mu(n/d) for 1 <= t <= n-1. (End)
EXAMPLE
The few first rows follow: c_n(t)
t 0 1 2 3 4 5 6 | t 1 2 3 4 5 6 7
n |n
1 1; |1 1;
2 1, -1; |2 -1, 1;
3 2, -1, -1; |3 -1, -1, 2;
4 2, 0, -2, 0; |4 0, -2, 0, 2;
5 4, -1, -1, -1, -1; |5 -1, -1, -1, -1, 4;
6 2, 1, -1, -2, -1, 1; |6 1, -1, -2, -1, 1, 2;
7 6, -1, -1, -1, -1, -1, -1; |7 -1, -1, -1, -1, -1, -1, 6;
... | ...
[Edited by Seiichi Manyama, Mar 05 2018]
MATHEMATICA
b[n_, m_] := b[n, m] = If[n > 1, b[n - 1, m] - b[n - 1, m - n + 1], 0]
b[1, m_] := b[1, m] = If[m == 0, 1, 0]
nt[n_, t_] := Round[(n - 1)/2 - t/n]
a[n_, t_] := Sum[b[n, k*n + t], {k, 0, nt[n, t]}]
Flatten[Table[Table[a[n, m], {m, 0, n - 1}], {n, 1, 20}]]
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Gevorg Hmayakyan, Feb 20 2017
STATUS
approved