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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


%S 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,

%T 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,

%U 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

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

%C The triangle T(n,k) contains many zeros. The distribution of nonzero entries is quite chaotic, but shows regular patterns, too, e.g.:

%C 1) T(n,1) > 0 for n prime or n=4; T(n,1)=0 else

%C 2) T(5k,k) > 0 for all k

%C More generally, it seems that:

%C 3) T(pk,k) > 0 for k>0 and primes p

%C The following table depicts the zero (-) and nonzero (x) entries for the first 80 rows of the triangle:

%C -

%C xx

%C x-x

%C xxxx

%C x---x

%C -xxxxx

%C x-----x

%C -xxx-xxx

%C --x-x-x-x

%C -x--xx--xx

%C x---------x

%C ---xxxxxxxxx

%C x-----------x

%C -x----xxxxxxxx

%C --x-x-x-x-x-x-x

%C -----xxx-x-x-xxx

%C x---------------x

%C -----x-xxx-x-x-xxx

%C x-----------------x

%C ---x---xxxxx-x-xxxxx

%C --x---x-x---x-x---x-x

%C -x--------xxxx----xxxx

%C x---------------------x

%C -------x-xxx-xxx-xxx-xxx

%C ----x---x---x---x---x---x

%C -x----------xx--xx--xx--xx

%C --------x-x-x-x-x-x-x-x-x-x

%C ---x-----x--xxxxxxxxxxxxxxxx

%C x---------------------------x

%C -----x---x-x--xxxxxxxxxxxxxxxx

%C x-----------------------------x

%C -------------xxx-x-x-x-x-x-x-xxx

%C --x-------x-x-x-------x-----x-x-x

%C -x--------------xx--------------xx

%C ----x-x---x---x-x-----x---x-x-x---x

%C -----------x-x-xxxxx---x-x-x-x-xxxxx

%C x-----------------------------------x

%C -x----------------xxxx------------xxxx

%C --x---------x-x---x-x-----x---x-x---x-x

%C -------x---x---x-xxx-xxx---x-x-x-xxx-xxx

%C x---------------------------------------x

%C -----x-----x-x-x-x-xxx-xxx---x-x-x-xxx-xxx

%C x-----------------------------------------x

%C ---x---------x------xxxxxxxx-x-x-x-xxxxxxxxx

%C --------x---x-x-x-x-x-x-x-x-x---x-x-x-x-x-x-x

%C -x--------------------xxxxxxxx--------xxxxxxxx

%C x---------------------------------------------x

%C ---------------x-x---xxx-x-x-xxx-x-x--xx-x-x-xxx

%C ------x-----x-----x-----x-----x-----x-----x-----x

%C ---------x---x---x---x--xx---x--xx---x--xx---x--xx

%C --x-------------x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x

%C ---x-----------x--------xxxx-x-xxxxx---xxxxx-x-xxxxx

%C x---------------------------------------------------x

%C -----------------x-x-x-x-xxxxx-x-xxxxx-x-xxxxx-x-xxxxx

%C ----x-----x---x---------x-----x---x---------x-----x---x

%C -------x-----x-----------xxx-xxx--xx-xxx-xxx-xxx-xxx-xxx

%C --x---------------x-x---------------x-x---------------x-x

%C -x--------------------------xx--xx--xx--xx--xx--xx--xx--xx

%C x---------------------------------------------------------x

%C -----------x---x---x-x-x----xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%C x-----------------------------------------------------------x

%C -x----------------------------xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%C --------x-----x-----x-x-x-x-----x-----x-x-x-x-----x-----x-x-x-x

%C -----------------------------xxx-x-x-x-x-x-x-x-x-x-x-x-x-x-x-xxx

%C ----x-------x---x---x---x---x---x---x---x---x-------x---x---x---x

%C -----x---------x-----x-x-x-x-x--xx-x---x-x---x-x-------x-x-x---xxx

%C x-----------------------------------------------------------------x

%C ---x---------------x------------xxxx-------------x-x------------xxxx

%C --x-------------------x-x-x-x-x-x-------x-x-x-x-x-x-------x-x-x-x-x-x

%C ---------x---x-x-x---x---x-x-x---xxxxx---x---x---x-x-x---x---x-x-xxxxx

%C x---------------------------------------------------------------------x

%C -----------------------x-x-x-x-x-xxx-xxx-x-x-x-x-x-x-x-x---x-x-x-xxx-xxx

%C x-----------------------------------------------------------------------x

%C -x----------------------------------xx--xx--------------------------xx--xx

%C --------------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

%C ---x-----------------x--------------xxxxxxxx---------x-x-x-x--------xxxxxxxx

%C ------x---x-----x-----x---x-x-----x-x---------x-----x---x-x-----x-x---x-----x

%C -----x-----------x-------x-x-x-x-x-x-xxxxxxxxx-x-x-x-x-x-x-x-x-x-x-x-xxxxxxxxx

%C x-----------------------------------------------------------------------------x

%C ---------------x---x---------------x-xxx-x-x-xxx---x---x-x-x-x-x---x-xxx-x-x-xxx

%C SUM(A057427(a(k)): 1<=k<=n) = A005127(n). - _Reinhard Zumkeller_, Jul 04 2009

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

%e Triangle starts:

%e 0;

%e 1, 1;

%e 2, 0, 1;

%e 2, 3, 2, 1;

%e 4, 0, 0, 0, 1;

%e 0, 4, 3, 1, 3, 1;

%e 6, 0, 0, 0, 0, 0, 1;

%e ....

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

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

%K nonn,tabl

%O 1,4

%A _Tilman Neumann_, Oct 04 2008, Oct 06 2008

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Last modified January 18 04:47 EST 2019. Contains 319269 sequences. (Running on oeis4.)