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A145201 Triangle read by rows: T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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: A079686 A005813 A049262 * A284265 A119464 A214568

Adjacent sequences:  A145198 A145199 A145200 * A145202 A145203 A145204

KEYWORD

nonn,tabl

AUTHOR

Tilman Neumann, Oct 04 2008, Oct 06 2008

STATUS

approved

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Last modified June 26 01:11 EDT 2017. Contains 288748 sequences.