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A123081
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Infinite square array read by antidiagonals: T(n,k) = Bell(n+k) = A000110(n+k).
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1
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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
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OFFSET
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0,4
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COMMENTS
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Alternatively, triangle read by rows in which row n (n >= 0) contains A000110(n) repeated n+1 times.
Row sums = A052887: 1, 2, 6, 20, 75, 312, ... A127568 = Q * M n-th row is composed of n+1 terms of A000110(n).
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LINKS
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FORMULA
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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; ...)
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EXAMPLE
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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;
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MATHEMATICA
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Table[BellB[n], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 21 2021 *)
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PROG
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(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
(Magma) [Bell(n): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2021
(Sage) flatten([[bell_number(n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 21 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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