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

1,4

Comments

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

Links

Table of n, a(n) for n=1..104.

Formula

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

Example

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;
....

Prog

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

Crossrefs

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

Sequence in context: A005813 A049262 A374748 * A323671 A340707 A284265

Adjacent sequences: A145198 A145199 A145200 * A145202 A145203 A145204

Keyword

nonn,tabl

Author

Tilman Neumann, Oct 04 2008, Oct 06 2008

Status

approved