A214810 - OEIS

%I #21 May 06 2024 14:37:22

%S 1,1,0,1,1,2,2,1,1,2,0,0,2,1,1,2,5,1,3,0,2,1,1,2,5,4,8,5,8,4,5,2,2,1,

%T 1,2,5,2,0,8,6,6,9,2,9,11,2,1,1,2,5,15,1,16,10,9,16,1,15,11,6,15,11,

%U 14,2,1,1,2,5,15,14,13,3,17,0,18,4,5,7,14,16,15,1,10,2,1,1,2,5,15,6,19,3,0,10,9,1,20,1,12,9,5,6,6,9,4,16,22,2,1,1,2,5,15,23,0

%N Triangle read by rows: T(n,k) (n>=1, 0 <= k <= p where p = n-th prime) = Bell(k) mod p (cf. A000110).

%C The n-th row gives Bell numbers mod prime(n) and has length prime(n)+1.

%H Alois P. Heinz, <a href="/A214810/b214810.txt">Rows n = 1..100, flattened</a>

%H J. Levine and R. E. Dalton, <a href="https://doi.org/10.1090/S0025-5718-1962-0148604-2">Minimum Periods, Modulo p, of First Order Bell Exponential Integrals</a>, Mathematics of Computation, 16 (1962), 416-423. See Table 2.

%e Triangle begins:

%e   [1, 1, 0],

%e   [1, 1, 2, 2],

%e   [1, 1, 2, 0, 0, 2],

%e   [1, 1, 2, 5, 1, 3, 0, 2],

%e   [1, 1, 2, 5, 4, 8, 5, 8, 4, 5, 2, 2],

%e   [1, 1, 2, 5, 2, 0, 8, 6, 6, 9, 2, 9, 11, 2],

%e   [1, 1, 2, 5, 15, 1, 16, 10, 9, 16, 1, 15, 11, 6, 15, 11, 14, 2],

%e   [1, 1, 2, 5, 15, 14, 13, 3, 17, 0, 18, 4, 5, 7, 14, 16, 15, 1, 10, 2],

%e   ...

%p T:= n-> (p-> seq(combinat[bell](k) mod p, k=0..p))(ithprime(n)):

%p seq(T(n), n=1..10);  # _Alois P. Heinz_, Jun 07 2023

%t A214810row[n_]:=Mod[BellB[Range[0,Prime[n]]],Prime[n]];Array[A214810row,50] (* _Paolo Xausa_, Aug 07 2023 *)

%Y Cf. A000110, A054767.

%K nonn,look,tabf

%O 1,6

%A _N. J. A. Sloane_, Jul 31 2012