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A350879
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Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is the number of partitions of n such that k*(greatest part) = (number of parts).
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9
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1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 3, 1, 1, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 1, 4, 1, 1, 1, 0, 0, 0, 0, 1, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1, 6, 3, 2, 1, 1, 0, 0, 0, 0, 0, 1, 7, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 11, 5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 11, 7, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,22
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COMMENTS
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T(n,k) is the number of partitions of n such that (greatest part) = k*(number of parts).
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LINKS
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FORMULA
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G.f. of column k: Sum_{i>=1} x^((k+1)*i-1) * Product_{j=1..i-1} (1-x^(k*i+j-1))/(1-x^j).
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EXAMPLE
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Triangle begins:
1;
0, 1;
1, 0, 1;
1, 0, 0, 1;
1, 1, 0, 0, 1;
1, 1, 0, 0, 0, 1;
3, 1, 1, 0, 0, 0, 1;
2, 2, 1, 0, 0, 0, 0, 1;
4, 1, 1, 1, 0, 0, 0, 0, 1;
4, 2, 1, 1, 0, 0, 0, 0, 0, 1;
6, 3, 2, 1, 1, 0, 0, 0, 0, 0, 1;
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PROG
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(PARI) T(n, k) = polcoef(sum(i=1, (n+1)\(k+1), x^((k+1)*i-1)*prod(j=1, i-1, (1-x^(k*i+j-1))/(1-x^j+x*O(x^n)))), n);
(Ruby)
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A(n)
a = Array.new(n, 0)
partition(n, 1, n).each{|ary|
(1..n).each{|i|
a[i - 1] += 1 if i * ary[0] == ary.size
}
}
a
end
(1..n).map{|i| A(i)}.flatten
end
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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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