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Infinite square array read by antidiagonals: T(n,k) = Bell(n+k) = A000110(n+k).
1

%I #21 Sep 08 2022 08:45:28

%S 1,1,1,2,2,2,5,5,5,5,15,15,15,15,15,52,52,52,52,52,52,203,203,203,203,

%T 203,203,203,877,877,877,877,877,877,877,877,4140,4140,4140,4140,4140,

%U 4140,4140,4140,4140,21147,21147,21147,21147,21147,21147,21147,21147,21147,21147,115975,115975,115975,115975

%N Infinite square array read by antidiagonals: T(n,k) = Bell(n+k) = A000110(n+k).

%C Alternatively, triangle read by rows in which row n (n >= 0) contains A000110(n) repeated n+1 times.

%C Row sums = A052887: 1, 2, 6, 20, 75, 312, ... A127568 = Q * M n-th row is composed of n+1 terms of A000110(n).

%H G. C. Greubel, <a href="/A123081/b123081.txt">Antidiagonal rows n = 0..50, flattened</a>

%H W. F. Lunnon et al., <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa35/aa3511.pdf">Arithmetic properties of Bell numbers to a composite modulus I</a>, Acta Arith., 35 (1979), 1-16. [From _N. J. A. Sloane_, Feb 07 2009]

%F M * Q, as infinite lower triangular matrices; M = the Bell sequence, A000110 in the main diagonal and the rest zeros. Q = (1; 1, 1; 1, 1, 1; ...)

%e Square array begins:

%e 1, 1, 2, 5, 15, 52, 203, 877, ...;

%e 1, 2, 5, 15, 52, 203, 877, 4140, ...;

%e 2, 5, 15, 52, 203, 877, 4140, 21147, ...;

%e 5, 15, 52, 203, 877, 4140, 21147, 115975, ...;

%e 15, 52, 203, 877, 4140, 21147, 115975, 678570, ...;

%e 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...;

%e 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, ...;

%e 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, ...;

%e First few rows of the triangle:

%e 1;

%e 1, 1;

%e 2, 2, 2;

%e 5, 5, 5, 5;

%e 15, 15, 15, 15, 15;

%e 52, 52, 52, 52, 52, 52;

%e 203, 203, 203, 203, 203, 203, 203;

%t Table[BellB[n], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 21 2021 *)

%o (PARI) B(n)=sum(k=0,n,stirling(n,k,2));

%o for(n=0,20,for(k=0,n,print1(B(n),", "))); \\ _Joerg Arndt_, Apr 21 2014

%o (Magma) [Bell(n): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 21 2021

%o (Sage) flatten([[bell_number(n) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 21 2021

%Y Cf. A000110, A052887, A127568.

%K nonn,easy,tabl

%O 0,4

%A _Gary W. Adamson_, Jan 19 2007

%E Edited by _N. J. A. Sloane_, Feb 07 2009

%E Added more terms, _Joerg Arndt_, Apr 21 2014