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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.

%I A145201 #16 Aug 10 2015 03:46:58
%S A145201 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 A145201 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 A145201 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 A145201 Triangle read by rows: T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.
%C A145201 The triangle T(n,k) contains many zeros. The distribution of nonzero entries is quite chaotic, but shows regular patterns, too, e.g.:
%C A145201 1) T(n,1) > 0 for n prime or n=4; T(n,1)=0 else
%C A145201 2) T(5k,k) > 0 for all k
%C A145201 More generally, it seems that:
%C A145201 3) T(pk,k) > 0 for k>0 and primes p
%C A145201 The following table depicts the zero (-) and nonzero (x) entries for the first 80 rows of the triangle:
%C A145201 -
%C A145201 xx
%C A145201 x-x
%C A145201 xxxx
%C A145201 x---x
%C A145201 -xxxxx
%C A145201 x-----x
%C A145201 -xxx-xxx
%C A145201 --x-x-x-x
%C A145201 -x--xx--xx
%C A145201 x---------x
%C A145201 ---xxxxxxxxx
%C A145201 x-----------x
%C A145201 -x----xxxxxxxx
%C A145201 --x-x-x-x-x-x-x
%C A145201 -----xxx-x-x-xxx
%C A145201 x---------------x
%C A145201 -----x-xxx-x-x-xxx
%C A145201 x-----------------x
%C A145201 ---x---xxxxx-x-xxxxx
%C A145201 --x---x-x---x-x---x-x
%C A145201 -x--------xxxx----xxxx
%C A145201 x---------------------x
%C A145201 -------x-xxx-xxx-xxx-xxx
%C A145201 ----x---x---x---x---x---x
%C A145201 -x----------xx--xx--xx--xx
%C A145201 --------x-x-x-x-x-x-x-x-x-x
%C A145201 ---x-----x--xxxxxxxxxxxxxxxx
%C A145201 x---------------------------x
%C A145201 -----x---x-x--xxxxxxxxxxxxxxxx
%C A145201 x-----------------------------x
%C A145201 -------------xxx-x-x-x-x-x-x-xxx
%C A145201 --x-------x-x-x-------x-----x-x-x
%C A145201 -x--------------xx--------------xx
%C A145201 ----x-x---x---x-x-----x---x-x-x---x
%C A145201 -----------x-x-xxxxx---x-x-x-x-xxxxx
%C A145201 x-----------------------------------x
%C A145201 -x----------------xxxx------------xxxx
%C A145201 --x---------x-x---x-x-----x---x-x---x-x
%C A145201 -------x---x---x-xxx-xxx---x-x-x-xxx-xxx
%C A145201 x---------------------------------------x
%C A145201 -----x-----x-x-x-x-xxx-xxx---x-x-x-xxx-xxx
%C A145201 x-----------------------------------------x
%C A145201 ---x---------x------xxxxxxxx-x-x-x-xxxxxxxxx
%C A145201 --------x---x-x-x-x-x-x-x-x-x---x-x-x-x-x-x-x
%C A145201 -x--------------------xxxxxxxx--------xxxxxxxx
%C A145201 x---------------------------------------------x
%C A145201 ---------------x-x---xxx-x-x-xxx-x-x--xx-x-x-xxx
%C A145201 ------x-----x-----x-----x-----x-----x-----x-----x
%C A145201 ---------x---x---x---x--xx---x--xx---x--xx---x--xx
%C A145201 --x-------------x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x
%C A145201 ---x-----------x--------xxxx-x-xxxxx---xxxxx-x-xxxxx
%C A145201 x---------------------------------------------------x
%C A145201 -----------------x-x-x-x-xxxxx-x-xxxxx-x-xxxxx-x-xxxxx
%C A145201 ----x-----x---x---------x-----x---x---------x-----x---x
%C A145201 -------x-----x-----------xxx-xxx--xx-xxx-xxx-xxx-xxx-xxx
%C A145201 --x---------------x-x---------------x-x---------------x-x
%C A145201 -x--------------------------xx--xx--xx--xx--xx--xx--xx--xx
%C A145201 x---------------------------------------------------------x
%C A145201 -----------x---x---x-x-x----xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%C A145201 x-----------------------------------------------------------x
%C A145201 -x----------------------------xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%C A145201 --------x-----x-----x-x-x-x-----x-----x-x-x-x-----x-----x-x-x-x
%C A145201 -----------------------------xxx-x-x-x-x-x-x-x-x-x-x-x-x-x-x-xxx
%C A145201 ----x-------x---x---x---x---x---x---x---x---x-------x---x---x---x
%C A145201 -----x---------x-----x-x-x-x-x--xx-x---x-x---x-x-------x-x-x---xxx
%C A145201 x-----------------------------------------------------------------x
%C A145201 ---x---------------x------------xxxx-------------x-x------------xxxx
%C A145201 --x-------------------x-x-x-x-x-x-------x-x-x-x-x-x-------x-x-x-x-x-x
%C A145201 ---------x---x-x-x---x---x-x-x---xxxxx---x---x---x-x-x---x---x-x-xxxxx
%C A145201 x---------------------------------------------------------------------x
%C A145201 -----------------------x-x-x-x-x-xxx-xxx-x-x-x-x-x-x-x-x---x-x-x-xxx-xxx
%C A145201 x-----------------------------------------------------------------------x
%C A145201 -x----------------------------------xx--xx--------------------------xx--xx
%C A145201 --------------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 A145201 ---x-----------------x--------------xxxxxxxx---------x-x-x-x--------xxxxxxxx
%C A145201 ------x---x-----x-----x---x-x-----x-x---------x-----x---x-x-----x-x---x-----x
%C A145201 -----x-----------x-------x-x-x-x-x-x-xxxxxxxxx-x-x-x-x-x-x-x-x-x-x-x-xxxxxxxxx
%C A145201 x-----------------------------------------------------------------------------x
%C A145201 ---------------x---x---------------x-xxx-x-x-xxx---x---x-x-x-x-x---x-xxx-x-x-xxx
%C A145201 SUM(A057427(a(k)): 1<=k<=n) = A005127(n). - _Reinhard Zumkeller_, Jul 04 2009
%F A145201 T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.
%e A145201 Triangle starts:
%e A145201 0;
%e A145201 1, 1;
%e A145201 2, 0, 1;
%e A145201 2, 3, 2, 1;
%e A145201 4, 0, 0, 0, 1;
%e A145201 0, 4, 3, 1, 3, 1;
%e A145201 6, 0, 0, 0, 0, 0, 1;
%e A145201 ....
%o A145201 (PARI) tabl(nn) = {for (n=1, nn, for (k=1, n, print1(stirling(n, k, 1) % n, ", ");); print(););} \\ _Michel Marcus_, Aug 10 2015
%Y A145201 Cf. A000040, A008275, A061006 (first column).
%K A145201 nonn,tabl
%O A145201 1,4
%A A145201 _Tilman Neumann_, Oct 04 2008, Oct 06 2008