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A213947
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Triangle read by rows: columns are finite differences of the INVERT transform of (1, 2, 3, ...) terms.
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2
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1, 1, 2, 1, 4, 3, 1, 10, 6, 4, 1, 20, 21, 8, 5, 1, 42, 57, 28, 10, 6, 1, 84, 150, 88, 35, 12, 7, 1, 170, 390, 252, 110, 42, 14, 8, 1, 340, 990, 712, 335, 132, 49, 16, 9, 1, 682, 2475, 1992, 975, 402, 154, 56, 18, 10
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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Create an array in which the n-th row is the output of the INVERT transform on the first n natural numbers followed by zeros:
1, 1, 1, 1, 1, 1, 1, ...
1, 3, 5, 11, 21, 43, 85, ... (A001045)
1, 3, 8, 17, 42, 100, 235, ... (A101822)
1, 3, 8, 21, 50, 128, 323, ...
...
For example, row 3 is the INVERT transform of (1, 2, 3, 0, 0, 0, ...). Then, take finite differences of column terms starting from the top; which become the rows of the triangle.
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LINKS
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EXAMPLE
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First few rows of the triangle:
1;
1, 2;
1, 4, 3;
1, 10, 6, 4;
1, 20, 21, 8, 5;
1, 42, 57, 28, 10, 6;
1, 84, 150, 88, 35, 12, 7;
1, 170, 390, 252, 110, 42, 14, 8;
1, 340, 990, 712, 335, 132, 49, 16, 9;
1, 682, 2475, 1992, 975, 402, 154, 56, 18, 10;
1, 1364, 6138, 5464, 2805, 1200, 469, 176, 63, 20, 11;
...
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MAPLE
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read("transforms") ;
A213947i := proc(n, k)
L := [seq(i, i=1..n), seq(0, i=0..k)] ;
INVERT(L) ;
op(k, %) ;
end proc:
if k = 1 then
1;
else
A213947i(k, n)-A213947i(k-1, n) ;
end if;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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