A005127 -id:A005127 - cp's OEIS frontend

A145201 Triangle read by rows: T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

0, 1, 1, 2, 0, 1, 2, 3, 2, 1, 4, 0, 0, 0, 1, 0, 4, 3, 1, 3, 1, 6, 0, 0, 0, 0, 0, 1, 0, 4, 4, 1, 0, 2, 4, 1, 0, 0, 8, 0, 3, 0, 6, 0, 1, 0, 6, 0, 0, 5, 3, 0, 0, 5, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 6, 11, 6, 3, 6, 5, 6, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 0, 0, 0, 0, 7, 5, 7, 7, 7, 7, 7

Offset: 1

Tilman Neumann, Oct 04 2008, Oct 06 2008

The triangle T(n,k) contains many zeros. The distribution of nonzero entries is quite chaotic, but shows regular patterns, too, e.g.:
1) T(n,1) > 0 for n prime or n=4; T(n,1)=0 else
2) T(5k,k) > 0 for all k
More generally, it seems that:
3) T(pk,k) > 0 for k>0 and primes p
The following table depicts the zero (-) and nonzero (x) entries for the first 80 rows of the triangle:
-
xx
x-x
xxxx
x---x
-xxxxx
x-----x
-xxx-xxx
--x-x-x-x
-x--xx--xx
x---------x
---xxxxxxxxx
x-----------x
-x----xxxxxxxx
--x-x-x-x-x-x-x
-----xxx-x-x-xxx
x---------------x
-----x-xxx-x-x-xxx
x-----------------x
---x---xxxxx-x-xxxxx
--x---x-x---x-x---x-x
-x--------xxxx----xxxx
x---------------------x
-------x-xxx-xxx-xxx-xxx
----x---x---x---x---x---x
-x----------xx--xx--xx--xx
--------x-x-x-x-x-x-x-x-x-x
---x-----x--xxxxxxxxxxxxxxxx
x---------------------------x
-----x---x-x--xxxxxxxxxxxxxxxx
x-----------------------------x
-------------xxx-x-x-x-x-x-x-xxx
--x-------x-x-x-------x-----x-x-x
-x--------------xx--------------xx
----x-x---x---x-x-----x---x-x-x---x
-----------x-x-xxxxx---x-x-x-x-xxxxx
x-----------------------------------x
-x----------------xxxx------------xxxx
--x---------x-x---x-x-----x---x-x---x-x
-------x---x---x-xxx-xxx---x-x-x-xxx-xxx
x---------------------------------------x
-----x-----x-x-x-x-xxx-xxx---x-x-x-xxx-xxx
x-----------------------------------------x
---x---------x------xxxxxxxx-x-x-x-xxxxxxxxx
--------x---x-x-x-x-x-x-x-x-x---x-x-x-x-x-x-x
-x--------------------xxxxxxxx--------xxxxxxxx
x---------------------------------------------x
---------------x-x---xxx-x-x-xxx-x-x--xx-x-x-xxx
------x-----x-----x-----x-----x-----x-----x-----x
---------x---x---x---x--xx---x--xx---x--xx---x--xx
--x-------------x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x
---x-----------x--------xxxx-x-xxxxx---xxxxx-x-xxxxx
x---------------------------------------------------x
-----------------x-x-x-x-xxxxx-x-xxxxx-x-xxxxx-x-xxxxx
----x-----x---x---------x-----x---x---------x-----x---x
-------x-----x-----------xxx-xxx--xx-xxx-xxx-xxx-xxx-xxx
--x---------------x-x---------------x-x---------------x-x
-x--------------------------xx--xx--xx--xx--xx--xx--xx--xx
x---------------------------------------------------------x
-----------x---x---x-x-x----xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x-----------------------------------------------------------x
-x----------------------------xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
--------x-----x-----x-x-x-x-----x-----x-x-x-x-----x-----x-x-x-x
-----------------------------xxx-x-x-x-x-x-x-x-x-x-x-x-x-x-x-xxx
----x-------x---x---x---x---x---x---x---x---x-------x---x---x---x
-----x---------x-----x-x-x-x-x--xx-x---x-x---x-x-------x-x-x---xxx
x-----------------------------------------------------------------x
---x---------------x------------xxxx-------------x-x------------xxxx
--x-------------------x-x-x-x-x-x-------x-x-x-x-x-x-------x-x-x-x-x-x
---------x---x-x-x---x---x-x-x---xxxxx---x---x---x-x-x---x---x-x-xxxxx
x---------------------------------------------------------------------x
-----------------------x-x-x-x-x-xxx-xxx-x-x-x-x-x-x-x-x---x-x-x-xxx-xxx
x-----------------------------------------------------------------------x
-x----------------------------------xx--xx--------------------------xx--xx
--------------x---x---x-x-x---x-x-x-x-x---x-x-x-x-x---x-x-x-x-x---x-x-x-x-x
---x-----------------x--------------xxxxxxxx---------x-x-x-x--------xxxxxxxx
------x---x-----x-----x---x-x-----x-x---------x-----x---x-x-----x-x---x-----x
-----x-----------x-------x-x-x-x-x-x-xxxxxxxxx-x-x-x-x-x-x-x-x-x-x-x-xxxxxxxxx
x-----------------------------------------------------------------------------x
---------------x---x---------------x-xxx-x-x-xxx---x---x-x-x-x-x---x-xxx-x-x-xxx
SUM(A057427(a(k)): 1<=k<=n) = A005127(n). - Reinhard Zumkeller, Jul 04 2009

			Triangle starts:
0;
1, 1;
2, 0, 1;
2, 3, 2, 1;
4, 0, 0, 0, 1;
0, 4, 3, 1, 3, 1;
6, 0, 0, 0, 0, 0, 1;
....

Cf. A000040, A008275, A061006 (first column).

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(stirling(n, k, 1) % n, ", ");); print(););} \\ Michel Marcus, Aug 10 2015

T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

cp's OEIS Frontend

A145201 Triangle read by rows: T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

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