A123081 Infinite square array read by antidiagonals: T(n,k) = Bell(n+k) = A000110(n+k).
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, 203, 203, 203, 877, 877, 877, 877, 877, 877, 877, 877, 4140, 4140, 4140, 4140, 4140, 4140, 4140, 4140, 4140, 21147, 21147, 21147, 21147, 21147, 21147, 21147, 21147, 21147, 21147, 115975, 115975, 115975, 115975
Offset: 0
Examples
Square array begins:
1, 1, 2, 5, 15, 52, 203, 877, ...;
1, 2, 5, 15, 52, 203, 877, 4140, ...;
2, 5, 15, 52, 203, 877, 4140, 21147, ...;
5, 15, 52, 203, 877, 4140, 21147, 115975, ...;
15, 52, 203, 877, 4140, 21147, 115975, 678570, ...;
52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...;
203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, ...;
877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, ...;
First few rows of the triangle:
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, 203, 203, 203;
Links
- G. C. Greubel, Antidiagonal rows n = 0..50, flattened
- W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16. [From _N. J. A. Sloane_, Feb 07 2009]
Programs
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Magma
[Bell(n): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2021
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Mathematica
Table[BellB[n], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 21 2021 *) -
PARI
B(n)=sum(k=0,n,stirling(n,k,2)); for(n=0,20,for(k=0,n,print1(B(n),", "))); \\ Joerg Arndt, Apr 21 2014
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Sage
flatten([[bell_number(n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 21 2021
Formula
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; ...)
Extensions
Edited by N. J. A. Sloane, Feb 07 2009
Added more terms, Joerg Arndt, Apr 21 2014
Comments