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A203300
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Self-generating triangle based on symmetric functions.
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2
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1, 1, 1, 1, 2, 1, 1, 4, 5, 2, 1, 12, 49, 78, 40, 1, 180, 11085, 270610, 2094264, 1834560, 1, 4210700, 4952544856489, 1094968722994345590, 11723079808649412379800, 2086231309557403469400000, 2074509324712524510720000
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,5
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COMMENTS
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Let row n+1 be (c0, c1, c2,...,cn). Then
c0*x^n + c1*x^(n-1) +...+ cn=(x+b0)(x+b1)...(x+bm),
where (b0,b1,b2,...,bm) is row n.
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LINKS
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FORMULA
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row n+1: f(0,r), f(1,r),...f(n,r), where f(k,r)=(k-th elementary symmetric function), r=(row n).
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EXAMPLE
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First five rows:
1
1...1
1...2....1
1...4....5....2
1...12...49...78...40
The factorization property is illustrated by
x^2 + 2x + 1 -> (x+1)(x+2)(x+1) = x^3 + 4x^2 + 5x + 2.
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MATHEMATICA
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s =.; s[1] = {1};
Prepend[Table[s[z] = Table[SymmetricPolynomial
[k, s[z - 1]], {k, 0, z - 1}], {z, 2, 7}], s[1]]
% // TableForm (* A203300 triangle *)
%% // Flatten (* A203300 sequence *)
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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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